Infinite Series in a History of Analysis: Stages up to the Verge of Summability 9783110359831, 9783110343724

Table of contents :
Foreword
Contents
1. The Greek era: Speculation, ideas and the comet of rigor
2. Successful experiments on infinite summation
3. Series of functions: the prototypes
4. Series seriously: the Greek comet reappears
5. On the verge of summability
Appendixes (of notes, proofs, exercises)
Life Data
References
Index

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Hans-Heinrich Körle Infinite Series in a History of Analysis De Gruyter Textbook

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Hans-Heinrich Körle

Infinite Series in a History of Analysis |

Stages up to the Verge of Summability

Author Dr. phil. habil. Hans-Heinrich Körle Marburg an der Lahn, Deutschland [email protected]

ISBN 978-3-11-034372-4 e-ISBN (PDF) 978-3-11-035983-1 e-ISBN (EPUB) 978-3-11-039916-5 Library of Congress Cataloging-in-Publication Data A CIP catalogue record for this book has been applied for at the Library of Congress. Bibliografische Information der Deutschen Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2015 Walter de Gruyter GmbH, Berlin/Boston Typesetting: PTP-Berlin Protago-TEX-Production GmbH, Berlin Printing and binding: CPI books GmbH, Leck ♾ Printed on acid-free paper Printed in Germany www.degruyter.com

| Meinen beiden Enkeln JAKOB und MORITZ

Foreword Imagine speedy Achilles chasing tortoises. Whenever he caught one, he couldn’t believe it. Or rather he shouldn’t. Being a Greek, he should have thought: Every time I get where the tortoise has been it is gone; hence, not any time I would close up. It cannot be thought – said an authority, the philosopher Zeno of Elea. A good two millenniums had to go by until a mathematician figured out that it happens. Thanks to infinite series … (By the way: No “millennia”, for I won’t write “formulae”.) The infinite, that means the infinitely small and, on the other hand, the infinitely large and many. In recent history, “infinitesimals” of space, time and matter took the place of the atoms which the ancient Greeks had puzzled over. In the end, their so-called “geometers” coped with the infinite. They evaded it through finite procedure and thus cultivated a virus whose outgrowth was to become mathematics’ main branch: analysis. Those Greek mathematicians were far away from the Calculus Newton and Leibniz created, way back – and ahead! The Greeks had mastered the infinite. With Newton, it is another sort of mastery. Infinite series were made his willing servants, i.e. series of functions which in fact had been employed before, in India. Soon they would develop a momentum of their own. Newton, Euler, Abel, Cauchy set milestones on the long march of power series, Euler and Fourier introduced trigonometric series. Except for Abel’s initiative, theirs needed to be clarified. Analysts like Dirichlet, Riemann, Dedekind, Weierstraß cared to consolidate analysis such that, eventually, one could count on infinite series: Calculus then justly applied to them and every convergent series would definitely be worth a value, a limit number that is. We report what was accomplished around 1900. Infinite series cannot be separated from general analysis. Their common history will be outlined here along significant contributions of the pioneers. There was a prodigy among them: Leonard Euler. Power series; to him they were the bulls to be taken by the horns. In an unorthodox fashion he made marvelous findings like the exponential function; in retrospect they proved correct. Euler had considered any series “valuable”, able to be valued if only properly formed. Indeed, divergent series were to become domesticated. By spotlighting, we will point out the beginnings of “limitation”, thus giving a glimpse on an overall theory of infinite series. The past may and must serve the present. Not in the least, I want to serve the needs of those who struggle with advanced calculus. To learn, in mathematics, is to comprehend. To learn the essentials of analysis, it pays off a lot to see how they developed in history: by trial and error, by necessity. Infinite series prove good teachers. Through extensive references, I tried to make background and sources available to everyone. Don’t mind if you don’t find because I didn’t remind myself – like not finding out when and where to have met the graphic which I chose for the cover’s emblem. * * *

VIII

Foreword

If you might welcome the book, then you are indebted to whom I owe it: To Professor Dr. Thomas Sonar from Tech.Univ.Braunschweig, without whose firm encouragement I hardly would have written it. I thank the publisher DE GRUYTER OLDENBOURG for their confidence in the author and feel much obliged to Dr. Gerhard Pappert of OLDENBOURG who gave me all, and more, support I could think of. He was engaged in the project on its first steps. The trouble with the last ones I shared, by harsh compromise, with the busy Project Editor, Mr. Leonardo Milla. I must thank for being granted my references. Ms. Siobhan Scannell, M.A., attended to the language control; Mrs. Nicole Schwarz, the graphic artist, and Mrs. Cornelia Horn did fine jobs on the figures outside and inside the book. Gratitude is due to my home institution, the Fachbereich Mathematik und Informatik at PhilippsUniversität Marburg. I have been well equipped and enjoyed professional help from our librarian, Mrs. Christa Seip, as well as from the EDP expert Mr. Burkhardt Fischer, grad.eng., who, ever ready for help, was welcome all the more as software is hardware to me, a red rag to an obstinate greenhorn. For the same reason, I highly appreciated assistance by Dr. Werner Liese: Again, I could readily produce any formula thanks to his version of LaTeX, by the name of LiTeX. Marburg, June 2015

Hans-Heinrich Körle

Contents Foreword

VII

1

The Greek era: Speculation, ideas and the comet of rigor

1

1.1

Infinity, challenging Greek philosophers ................................................................... 1

1.2

Infinity, mastered by a Greek “geometer”: Eudoxus.................................................. 2

1.3

Archimedes and the geometric series to be ................................................................ 5

1.4

Memorizing a joint venture........................................................................................ 9

2

Successful experiments on infinite summation

2.1

Oresmus and the geometric series............................................................................ 11

2.2

Freestyle: Cavalieri summing indivisibles ............................................................... 13

2.3

Leibniz’ telescopes ................................................................................................... 16

3

Series of functions: the prototypes

3.1 3.1.1 3.1.2

Preliminary: Aspects ................................................................................................ 19 Series versus sequences ........................................................................................... 19 Series: evaluation and expansion ............................................................................. 20

3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6

The power of power series ....................................................................................... 23 Nīlakantha and the arcustangent series .................................................................... 23 The hyperbolic logarithm and its series ................................................................... 26 Newton and the binomial series ............................................................................... 29 Taylor series and expansion; ordinary power series................................................. 32 The binomial series in the hands of Euler ................................................................ 36 Euler’s algebraical analysis: evaluation and valuation............................................. 42

3.3 3.3.1 3.3.2 3.3.3 3.3.4

The power of trigonometric series ........................................................................... 46 Facing antipodes ...................................................................................................... 46 Euler evaluating a trigonometric series .................................................................... 46 Priming problems ..................................................................................................... 49 The mysterious . . . Fourier series ............................................................................ 50

3.4 3.4.1 3.4.2

Cauchy ..................................................................................................................... 55 Series of powers in the hands of Cauchy and Laurent ............................................. 55 The Cauchy product ................................................................................................. 61

11

19

X

Contents

4

Series seriously: the Greek comet reappears

67

4.1

Rigor in retrospect and prospect ...............................................................................67

4.2 4.2.1 4.2.2 4.2.3

Farewell to the infinitely small; the “epsilontics”.....................................................69 Convergence, continuity ...........................................................................................69 Dirichlet, Heine: uniform continuity ........................................................................71 From Abel to Weierstraß: uniform convergence .......................................................72

4.3

Welcome to irrationals: the complete space of real numbers ....................................78

4.4

Another complete space: the home of convergent sequences ...................................84

5

On the verge of summability

5.1

Divergent series: suspected and respected ................................................................89

5.2 5.2.1 5.2.2 5.2.3 5.2.4

The initiation of Cesàro and Abel summation ..........................................................90 Grandi’s series, recurrent series ................................................................................90 Frobenius’ theorem and the limit theorems of Cauchy and Abel ..............................92 The Cauchy product revisited by Abel and Cesàro ...................................................94 Introducing the methods of basic Cesàro and of Abel means ..................................95

5.3 5.3.1 5.3.2 5.3.3 5.3.4 5.3.5 5.3.6

Features of Cesàro and Abel means ..........................................................................98 Inclusion, limitation, efficiency ................................................................................98 The C1 limit: a continuous functional on a Banach space ......................................101 An early triumph of arithmetic means: Fejér’s Theorem ........................................103 On the scale of Cesàro means .................................................................................103 Cesàro’s Cauchy product ........................................................................................106 Inverse theorems, Tauber’s theorem .......................................................................107

89

Appendixes (of notes, proofs, exercises)

111

Life Data

119

References

121

Index

127

1

The Greek era: Speculation, ideas and the comet of rigor

1.1

Infinity, challenging Greek philosophers

Eternity, infinity. Sort of ciphers to label some-“thing” which is grounded in sentiment rather than sense. I feel ill at ease to think a procedure without end, halving no easier to imagine than is doubling. A bit like the sensation of vertigo that hits me when, walking under a starry night, I am haunted by the question why matter should exist at all. Whenever man enjoyed the spare time to think beyond physical survival, his mind started wondering, longing for apprehension. Some of his species would not rest content with irrational answers. To us, the home of rational thought was ancient Greece. Her early thinkers would ask how things might, and could at all, happen in space and time. They were known as sophists, men of wisdom. One in high repute was Zeno (Zenon) of Elea.{1a, 2a} The negative connotation of today’s sophism is due to people subsequent to him who, purposefully or not, set traps for the human mind and found caught themselves. Zeno wanted to make his countrymen think, when he made Achilles race a tortoise with a head start.{2b} He did not ask when Achilles would close up to it, but whether. No he said, never. This is why{1b, 3a}: Both start at t0 o’clock. Achilles starts from P0 and reaches, at t1, the starting position P1 of the tortoise who then had gained P2; Achilles gets to P2 by the time t2, and so on and on. The time scale t0 < t1 < t2 < ... never ends, so what about the time of race? Zeno presumably was facing this paradox: Infinitely many periods need be added up – to whatever else but to infinity!? So, what is time if it refutes thought? The sophist touched infinite series and their problems. It took more than two millenniums for an answer and Grégoire de Saint-Vincent{1c, 2c, 3b} with another kind of thought. He found out that the periods tn+1 – tn, n = 0,1, … , form a geometric series{2d, 3c} the convergence of which he visualized similarly as shows the book’s cover (see 1.3). Sophistication at its best when this contemporary of de Fermat resolved the Greek’s riddle by a mathematical model! (The answer was questioned by a certain Sir{4}; see Appx. A-1.) Facing the time required to cover a distance, Zeno presumably would have had to deny any move to be possible. Would he?{1d} How could an arrow fly when standing still every moment?{2e} Motion was visible, yet not thinkable. He was not the only “Zeno”. The Greeks, yes, a whole people’s mind set up. Eventually, their men of wisdom saw but one way out: Achilles would fail on account of the presumption that time and space could be subdivided open end.{2f} Partition was made a point by another philosopher when Greek geometry began to measure its objects. Democritus (Demokritos){5a} had the idea that all matter is composed by indivisibles, “atoms” in Greek. Did he think of the material world only? The objects of ideal geometry, ought

2

1 The Greek era: Speculation, ideas and the comet of rigor

they not be partitioned as ever one wants? In fact, geometrical atoms were discussed seriously. Democritus was about to imagine a cone to be made up of layers parallel to the bottom; if those layers of atoms have or have no thickness, either assumption wouldn’t work.{5b} Is there something in between? To fill the gap, philosophers would resort to qualities like infinitely flat or slim or short – without being nothing! It meant an indivisible to own some dummy volume or area or length capable to create something proper. Really a dummy solution, just words. Yes? Though being absurd and questioned by Democritus himself{5b}, the idea did lead to correct conclusions, thus turning out two-edged. The pros and cons also startled Aristotle (Aristoteles).{3c} It was the infinite small rather than its big brother that kept philosophers busy. Aristotle would admit to the endless process so as to substantiate a potential infinite. However, later on at least, he definitely refuted the actual existence of an infinite aggregate like “the entity” formed by all the numbers 1,2, ... , called an actual infinite{6}. It would apply to infinitely small elements in geometry; by his own geometrical argument{3c}, Aristotle disproved the line elements of his time. Modern analysis will bring together what those philosophers must have had a feeling about: the infinitely small in size and the infinitely large in number might join, thus resulting in something finite. Euler was to be most successful when he made this a principle (3.2.5). As far as mathematics is involved, the paradoxes of Democritus and Zeno have been resolved by our concepts of integration and infinite series. {1} {2} {3} {4} {5} {6}

CAJORI [1]: {1a} 23. {1b} 23 (bottom). {1c} 181. {1d} 24. SONAR: {2a} 48. {2b} 54. {2c} 224. {2d} 225–227. {2e} 55. {2f} 56 below. KÖRLE: {3a} 13 f. {3b} 41. {3c} 14. DAMPIER: {4a} [1] 20 lines 13–15; [2] 28 lines 15–19. {4b} [1] 462 f; [2] 544 top. VAN DER WAERDEN: {5a} [1] 106, 137; [2] 176, 226. {5b} [1] 138; [2] 228. KLINE 992 bottom.

1.2

Infinity, mastered by a Greek “geometer”: Eudoxus

The area of a triangle reads A=

1 2

b hb ,

(1)

where b measures one of its bases and h is the respective height; geometrical evidence assures the value not to depend on the choice of base. (As usual, geometric objects and their measures will be denoted the same.) Our youngsters in school are well used to (1), Greek geometers were not, the formula being “alien to Euclid” (Eukleides).{1} To measure the triangle they rather asked for a square equal in size (Appx. A-2). So far, as the simplex of dimension 2 is concerned. In dimension 3, the simplex is the triangular pyramid or tetrahedron, made up of four triangles. How does its volume V result from a basic area B and the respective height hB? The formula analogue to (1) reads V=

1 3

B hB ,

(2)

1.2 Infinity, mastered by a Greek “geometer”: Eudoxus

3

and in fact, it’s true! However – on quite different grounds! Euclidean geometry provides (1). Since the sides of those simplexes are linear or plane, respectively, Euclid might be expected to likewise do the job in case of (2). He didn’t! (See below.) * * * The formulas (1), (2) are the way we communicate mathematics; the letters stand for numbers. We do so for convenience when talking about Greek mathematics. It is geometry and is written in geometry. To the learned [read learn-éd] Greeks numbers were the integers 2,3, ... , exclusively.{2a} Except for Euclid’s “theory of numbers” where they could multiply each other oftentimes, numbers in the first place meant segments of the straight line, whence the distributive law was given in terms of rectangles{2b} and multiplication could at most result in cuboids{2c}. Only in the end of the great era, arithmetic found its way from the merchants to mathematicians. * * * The history of (2) begins with those huge monuments at the Nile River. Such a right squarebased pyramid was estimated to measure one third the volume of the cuboid that shares base and height with it. When touring Egypt, Democritus presumably learned about that rule.{2a} According to what Archimedes{2a} tells us, Democritus asked for the precise ratio and made his correct answer touch the bottom of the problem. Democritus considered the Nile pyramid together with its circumscribed cuboid while, according to Fig. 1.1{2a, 3a}, quartered into congruent tetrahedrons and their circumscribed prisms. With reference to the prism P = ABMCDT, the problem of his pyramid reduces to show that the tetrahedron T = ABMT measures one third of P. It is true if T is equal in size to the other two tetrahedrons inside the prism P, which Democritus is said to have grounded on the following fundamental Proposition. Tetrahedrons of equal bases and heights are equal in size. Why does this account for the trisection? In the first place we note that T = CDTB [with common height MT]; secondly, we choose AMT for the base of T and get T (= ATMB ) = ATCB [with common height MB] (Appx. A-3). That Proposition was among the first challenges of analysis that were mastered, its proof being a masterpiece of Eudoxus (Eudoxos) from Cnidos.{2b} He was a friend, disciple and teacher of Plato [Platon, in Greek].

C

T

D

B

A M

Fig. 1.1. Trisection of the prism.

4

1 The Greek era: Speculation, ideas and the comet of rigor

Let us first consider how Democritus will have approached the Proposition on grounds of the philosophy he is known for.{4} Our formulas (1), (2) respectively say that, given a base b or B, the area or volume of the figure would not change as its top vertex is shifted along a straight line parallel to b or within a plane parallel to B. Democritus would imagine the said figures to be made up of infinitesimal linear or planar elements parallel to that specific base and being loosely adhesive so as to follow the movement of the top.{3b} (Later, the idea will form Cavalieri’s principle; see 2.2.) Fig. 1.1 shows a right prism. The argument that is ascribed to Democritus would apply to any prism partitioned in this way (Appx. A-4). * * * A proper proof was waiting for another species of “geometers”. Greece’s philosophers had done their job. They made people think, showed them the fallacies of thinking. Eudoxus and Archimedes, Greece’s foremost mathematicians, were both conscious of this when facing the problem of quantity in geometry. Apparently on the present occasion, the former conceived a revolutionary idea of proof which, by some writer{3c} in the 17th century, was named “exhaustion” {2c, 3d} – “unhappily” {2c}. And here is how it applies to two tetrahedrons with equal bases and heights, covered by {2d}. Prisms of equal bases and heights have the same volume;{2e} this holds by Euclidean geometry, to be quarried out of Euclid’s “Book XI”{5a} rather than quoted from it. The analogue for tetrahedrons{5b}, due to Eudoxus{2d}, is organized by a sequence of steps where prisms are simultaneously removed from two tetrahedrons (these and parts thereof are denoted by their volumes). Step 1: Take from V ', V " a set of prisms each such that their respective volumes amount to the same value 1

1

1

1

υ1 > 2 V ', > 2 V " , leaving rests R'1 ≔ V ' – υ1 < 2 V ' , R"1 = V " – υ1 < 2 V ". Step 2: Take from R'1 , R"1 a set of prisms each such that their respective volumes amount to the same value 1

1

1

1

υ2 > 2 R'1 , > 2 R"1 , leaving rests R'2 ≔ R'1 – υ2 < 2 R'1 , R"2 = R"1 – υ2 < 2 R"2 . 1

1

1

And so forth. First we note R'n < 2 R'n–1 < … < ( 2 )n V ' and likewise R"n < ( 2 )n V " so that the R'n , R"n become arbitrarily small. As V ' = R'1 + υ1 = R'2 + υ2 + υ1 = … and the like, we get V ' = R'n +

Σnk=1 υk ,

V " = R"n +

Σnk=1 υk

(3)

and thus |V ' – V "| = |R'n – R"n| ≤ R'n + R"n which shows the assumption 0 < |V ' – V "| disproved. * * * The Wonder-of-the-World on the banks of the Nile stand for a milestone in human history. In our story it became a founding-stone, even in the present subject matter: When reading the above proof that culminates in (3) we understand the volumes υ1, υ2 , ... to constitute an infi-

1.3 Archimedes and the geometric series to be

5

nite series that – implicitly – is verified convergent to V ' = V ". It is an early, if not the first, occasion where our series virtually come into being. Eudoxus precisely argued on a problem which only recently could be attacked in a like manner of precision. By taking the infinitely long route, he had left the path of geometry. Must he? This was brought into focus{6a} after polygons were proven equal in area if and only if they are “dissectionally congruent”, i.e. can be dissected into polygons P1, ... , Pn and Q1, ... , Qn such that any Pν is congruent to Qν (cf.{3b}). Could polyhedrons likewise qualify to have equal volumes? Gauß had doubts about it, David Hilbert made it no.3{7a} of those 23 problems{7b} he released on the threshold to the 20th century. The answer came the same year: no; tetrahedrons of equal volumes need not be dissectionally congruent (“raumgleich”){6b, 8}. So, formula (2) belongs to the “analysis of the infinite” as Euler would say. * * * Eudoxus made those indivisible and infinitely small elements go. Forever? They were to prove long-lived. Even under their name (see 2.2) which, on the eve of Calculus, changed into infinitesimals. The German Infinitesimalrechnung is “calculus of infinitesimals”. When analysis experienced Enlightenment, this denotation became a symbolic one. Not long ago, somebody made it concrete again. These very infinitesimals were reanimated to play a legal role in a “non-standard analysis”. {1} {2}

{3} {4} {5} {6} {7} {8}

1.3

CAJORI [1] 33. VAN DER WAERDEN: {2a} [1] 108, 265 f; [2] 180, 439f. {2b} [1] 118; [2] 194. {2c} [1] 119; [2] 195f. {2d} [1] 137 f; [2] 227 (at bottom, read ein dreiseitiges Prisma ... zerlegt werden). {2e} [1] 179 ff; [2] 292 ff. {2f} [1] 184; [2] 304. {2g} [1] 186 f; [2] 307 f. {2h} [1] 186 bottom; [2] 308. KÖRLE: {3a} 101. {3b} 102. {3c} 17. {3d} 17, 103 (at bottom, read Tetraeder, not Prisma). SONAR 52. EUKLID: {5a} 315–353; 343 § 323; 467, for “XI, 24” read “XI, 25”. {5b} 357–362. STILLWELL: {6a} 58 above. {6b} 58 below. HILBERT: {7a} 301 f. {7b} 290–329. DEHN 354.

Archimedes and the geometric series to be

We solve the quadrature of the triangle by formula 1.2(1), the ancient Greeks solved Appx. A-2. They were despairing of a problem which became proverbial, Plato’s quest for the quadrature of the circle, i.e. for the square that is equal to a given circle and bound to be constructed by (unmarked) ruler and compass exclusively. As incentive a challenge as a disastrous one. With great zeal and more or less seriously, the solution of the problem kept being searched for until, two millenniums later, the key was found … to open an empty box.

6

1 The Greek era: Speculation, ideas and the comet of rigor

After all, it is the story of a veritable progress. When Plato invited the world to solve his problem he presumably left open whether it was possible at all. Aside from the construction, he would suppose an equal square to exist in the Platonic sense. Philosophers are not used to leaving a thesis without an antithesis. Some were ruling that something crooked could not equal in size to something even. It was young Archimedes who was to disprove the dogma by a striking discovery. Out of 1.2(3) there developed an infinite series whose terms were not explicit. Archimedes was forming a “series” of areas that follow geometric digression. To him, open end summation does not result in a “sum total”. Yet he in fact evaluated a geometric series when he proved what in our terms includes 1

∫ 0 x2 dx

=

1 k Σ∞ ( ) k=1 4

1

= 3.

(1)

The “paper” Archimedes wrote is a letter{1, 2a, 3a} where he tells a friend that he had “squared the parabola”, which is to say its segments. To convey his idea we will reduce the subject so as to link with (1). However, to better appreciate the achievement, we take a total view beforehand. * * * Archimedes refers to a segment SAB under any chord AB.{2b, 3b, 4a, 5a, 6a} (In Fig. 1.2, the segment contains the vertex of the parabola like on {2c, 3c}.) He knows how to construct the tangent parallel to AB{2c, 3c}, its point P of contact forms the triangle ABP. Archimedes proves that ABP compares in size to SAB by ratio, namely by three to four.{2d, 3d} So, the segment can even be squared complying with Plato’s demand (see and work on Appx. A-5). B

A

Y2

X2 Y1

X1

P

T

Fig. 1.2. A parabolic segment is by one third greater than the triangle with the same base and height.

The master had studied the parabola thoroughly, including mechanical heuristics (cf. 1.4) as concerns the segment’s area.{2e, 3e} The proof makes use of “coordinate points” X,Y (Fig. 1.2). Choose the points X on the curve to the right of the straight line through P which, if not equal, is parallel to the (dotted) axis of symmetry; have Y on that line assigned to X according as XY||AB. With |PY| denoting the length of PY and with Q(X,Y) the area of the square over XY, the law of the parabola reads |PYi| : |PYk| = Q(Xi,Yi) : Q(Xk,Yk),{5b} meaningful thanks to Eudoxus’ theory of proportions.{4b} (We would write |PY| = a|XY|2 with a constant a.)

1.3 Archimedes and the geometric series to be

7

Archimedes’ proof by so-called exhaustion is sketched with reference to the parabolic segment SAB in Fig. 1.3. For more details see {4a, 5a, 6a}. B

B

C P** P

P A

P*

A

Q 3/4

Fig. 1.3. The first two steps of the parabolic segment’s exhaustion.

D 1 1

1

1

3

9

With P( 2 | 4 ), P*( 4 | 16 ), P**( 4 | 16 ), we note A0 ≔ | ABP | =

1 2

1

1

1

| ABCD | = 2 ( 4 · 1) = 8 , 1

1

1

1

A1 ≔ | APP*| + |PBP**| = 2 | APP*| = 2 2 ( 16 · 2 ) = 32 ; this likewise continues ad infinitum. Archimedes ascertains Ak−1 : Ak = Ak : Ak+1 , k = 1,2, ... . (It says that each area A1, A2, ... is the mean proportional or geometric mean Ak = √(Ak–1 Ak+1) of its neighbors and accounts for the denotations geometrical progression and series (cf. Appx. A-6). Therefore Ak Ak–1 A2 A1 A1 k Ak A0 = Ak–1 · Ak–2 ·…· A1 · A0 = ( A0 ) , k = (1,) 2,3, … ,

(2)

the finite sums of which were then formed the Archimedean way{4c, 5c}, in our terms

Σnk=0 Ak

= A0 Σk=0 ( 4 ) = A0[1 − ( 4 ) n

1 k

1 n+1

]/(1 − 14 ) .

(3) 4

To us, the job would thus be done, resulting in the limit 3A0 as n → ∞ in accord with (1), i.e. 1

∫0 x2 dx =

1 2

– |SAB| =

1 2

∞

– Σk=0 Ak =

1 2

–

1 4 · 8 3

1

=3.

Archimedes had to devise by himself the “calculus” he needed. To him, the sum formula in (3) was but the means for approaching the provisional value 4/3 as close as wanted – that meant nothing else but our technique with “ ε”. He was aware of “convergence” the very way we understand it, being sensitive up to a most subtle detail: For (¼)n to become arbitrarily small, Archimedes points out a postulate to be required while rather tacitly used by his contemporaries. He attributes it to Eudoxus, later it is called the Lemma of Archimedes{4d, 6b} and provides our Archimedean Axiom for real numbers. His demonstrations are as rigorous

8

1 The Greek era: Speculation, ideas and the comet of rigor

as it could be today; van der Waerden puts it thus: “… the ‘epsilontics’ … were for Archimedes an open book. In this respect, his thinking is entirely modern.” {4e} * * * As to Archimedes the mathematician, a result must be the result of proof. Yet, he liked to approach them any way, not for the sake of heuristics alone. Well, let us speculate on how Archimedes might have looked for a straightforward access to the present result, how he might have seen the sums converge. Archimedes the magician was a master of simulation (cf. 1.4). It wouldn’t look strange watching him simulate planar quantities by linear ones and interpret A0 : A1 = A1 : A2 = … above as relating linear segments such as to visualize convergence of A0 + A1 + A2 +… to a final linear segment that measures the area of the parabola’s segment. Exactly this concept was performed by the time when the Frenchman de Fermat and the Belgian de Saint-Vincent dealt with geometrical series (cf. 1.1). Through Fig. 1.4a, which is like Saint-Vincent’s diagram {7}, the argument is immediate by those night-caps’ similarity. Hard to believe that Archimedes might have resisted scratching something like Fig. 1.4a in the sand. If, then he would have quickly erased it…

A0 + A 1 + A2 + …

Fig. 1.4a. The geometric series by Saint-Vincent.

The author of that figure saw it at once serving to resolve Zeno’s paradoxon (cf. 1.1). Let any An, n = 0,1, … , be identified with the period tn+1 – tn of time by which Achilles gets from Pn to Pn+1. Thus, with velocities C > c, the length of the segment PnPn+1 is represented by CAn and cAn–1 which results in An : An–1 = c : C, n = 1,2, … , hence a geometric progression with a convergent “sum”. (Of course, the segments A0, A1, … may represent the stages of Achilles’ trip, yet not “as well” since Zeno’s problem would be missed that way.) The Greeks were thinking in terms of geometry and so Archimedes was unlikely to resolve Zeno’s paradox of time.

x2 x

s

1 s –1 1

Fig. 1.4b. 1 + x1 + x2 + … = s, where x : 1 = (s – 1) : s.

Saint-Vincent’s triangles became modified in Fig. 1.4b. It covers the book to catch the freshman’s eye, make him experience convergence. In fact, it addresses the two aspects of

1.4 Memorizing a joint venture

9

infinite series (cf. 3.1.2), namely evaluation and expansion: so to speak the finite composite of the infinite and the infinite decomposition of the finite. For good reason, Archimedes was proud of his accomplishment. Indeed, the geometric series was to become a leitmotiv in analysis, a leading motif. (It will be discovered many times.) {1} {2} {3} {4}

{5} {6} {7}

1.4

HEATH 29 f. ARCHIMEDES’ WERKE / HEATH: {2a} 354f. {2b} 354–370. {2c} 355 Satz 1. {2d} 365 Satz 17. {2e} 358ff. ARCHIMEDES · WERKE / CWALINA: {3a} 153. {3b}153–176. {3c} 154 §1. {3d} 168 §17. {3e} 156ff. VAN DER WAERDEN: {4a} [1] 216–220; [2] 361–366. {4b} [1] 187; [2] 286 f. {4c} [1] 219; [2] 365f. {4d} [1] 178, 186, 220f; [2] 290, 306 bottom, 367f. {4e} [1] 220; [2] 367. EDWARDS: {5a} 35–39 (figures 7, 8 need revision; cf.{2c, 3c}). {5b} 37(5). {5c} 39(7). KÖRLE: {6a} 111–115. {6b} 120f. SONAR: 227, Abb. 5.5.21.

Memorizing a joint venture

Eudoxus and Archimedes left an “analytical geometry” to the world (much different from today’s analytic geometry). They performed geometry by genuine analysis. It is not in memoriam when we look back on their ever surviving work, a joint venture though our protagonists differ by more than half a century. They translated into mathematics what was initiated by Democritus. When spelled in our language, in terms of infinite series that is, Eudoxus proved two of these to equal in value (1.2(3)) and Archimedes evaluated one by rational value (1.3 (3)). Euclid’s book covers what we owe to Eudoxus (1.2{5}, e.g.). Having left no written evidence of his own, he remained standing in the shadow of Archimedes. Eudoxus had become the creative theoretician of Greek mathematics, the latter excelled as the ingenious executive. When, in the course of an exhaustion process, Archimedes hit upon what we call the partial sums 1.3(3) of an infinite series he clearly saw the way the cat is jumping. However, he strictly followed the path Eudoxus had traced out which then was the only one to substantiate rigor outside the range of finite mathematics. There, proof could but be performed by disproving the contrary. Greek rigor had its price. In order to know what to prove, Archimedes had to be inventive, each time anew. He cultivated heuristics, the art of finding, and he never took it for proving. To that end, his mechanical lever became instrumental for comparison: volumes, areas and lengths were simulated by weight (Latin simul is like). The lever’s law was applied piecewise, thus reminding of, yet not adopting, the structure Democritus had introduced. As to volumes, homogeneous solids were assumed to split up into parallel planar sections allowing the substance to be weighed out slice by slice, areas being treated accordingly by straight line segments.{1a} Archimedes himself detailed this technique within a specific text which – a true sensation – was discovered right after 1900 and named “The Method” in literature{1} since it points out

10

1 The Greek era: Speculation, ideas and the comet of rigor

access to a value before proving the value. This “guide” had its première{2a} when Archimedes prepared to square the parabola (1.3): “... the very first theorem that became clear to me through mechanics”{2b}. For a typical example see {3}. There was system to what he did, yet no drive for generalization, a substantial trait of modern mathematics where progress is largely due to unifying concepts. The master’s strategy worked perfectly and he felt no need to further justify any experiment. He touched the infinite, but never got his hands stung with it. The culture of proof which marked the Golden Age of Greek mathematics was to break off right after Archimedes. Even the last of those Greek geometers, Apollonius of Perga, did not take it up, let alone the Romans. It was a long way to go for a renaissance comparable to Greek standards. Most pioneers of infinitesimal calculus will frankly seize upon the archaic idea of decomposition and turn the heuristic tool of Archimedes into a means of argument. Those subtle elements, after having been called atoms or indivisibles, were renamed into our key-word, the infinitesimals. (For a modern convention of distinction see {4}.) They provided the naïve concept that underlay the definite integral by the time calculus was ready for birth: an “infinite sum of infinitely small differences”. Folklore, living on in Leibniz’ suggestive symbols “∫”, “d” for summa and differentia. Archimedes had plenty of interests and skills; in mathematics, his tremendous production was a paradigm of precision and also brilliant by the clearness of its presentation. In these regards he compares with Euler. Archimedes became known during Renaissance, whereas the ideas of Eudoxus were given the attention they deserved no sooner than in the 19th century, and rather late after they had been developed over again. Greek philosophers had played their part in our story in that they pointed out the impasses. In the end, their commitment would become a school of mind with mathematical thought its precept – the renaissance of which was to form an essential of our culture. As before, mathematics would not prove that stupid doing as being maintained by our philosopher Arthur Schopenhauer who expressly refers to the analysis finitorum et infinitorum.{5} {1} {2} {3} {4} {5}

{1a} HEIBERG; ZEUTHEN 322–342. ARCHIMEDES · WERKE / CWALINA 383. {1b} GREEK MATHEMATICS II, 220 f. {2a} GREEK MATHEMATICS II 223. {2b} VAN DER WAERDEN [1] 213 f; 216–220; [2] 354356; 361–366. HEIBERG; ZEUTHEN 325 f. GOULD 473 f. KÖRLE 122 f. SONAR 52 (Abb. 2.2.3). HEUSER [2] 701. KÖRLE 99.

2

Successful experiments on infinite summation

Greek tradition had to find its way via the Orient. Arabs assimilated what they found in Greek papyri and made remarkable contributions. All this became known in the Occident’s late Middle Ages through Latin translations from Greek and Arabian. In the West, Fibonacci from Pisa [Pisanus] was maybe the first and single one to seriously deal with mathematics in the 14th century. He described a sequence that was to become most famous: the Fibonacci numbers. They diverge.

2.1

Oresmus and the geometric series

Convergence must appear somehow. Whenever a limit was at stake, Archimedes could impeccably ascertain existence on his terms. Far remote from that precision, some people set out to engage in infinite procedures again. They centered at Oxford and Paris. Oxford was the place of Roger Bacon{1, 2a, 3a, 4a}, a clergyman of the 13th century who held the view that nature should no longer be subject to some degenerate philosophy, but rather become the object of unbiased study, supported by mathematics wherever possible.{2b} He paid for it by incarceration, since by his time European thinking was governed by the doctrines of an Almighty Church and, as far as compatible, by those of Aristotle. A French bishop by the name of Nicole Oresme or d’Oresme [ɔr'ɛ:m], alias Nicolas Oresmus{3b}, seized upon the ideas of the Oxford school. There had been inquiry into dummy motions, mental experiments that transgressed reality not for the sake of sophistical speculation, but to dress serious mathematics in a spectacular fashion. Oresme strikingly improved upon the verbal solutions from Oxford. In the course of this, he visualized convergence and value of a geometric series as follows.{2c, 3c} On the left hand of Fig. 2.1{5a} areas in geometric digression are piled up to infinity, rectangles that Oresme thus rearranges: He shifts "½" to the left, drops "¼" down onto square "1" and shifts it left to join "½", and so forth. This results in ∞ 1 Σk=1 (2)

k

= 1.

(1)

Oresme trusted in what he saw; “… , vidi, vici” Caesar said, “I … saw and so I won.” When Oresme said “it’s obvious”, he was to be followed by quite a spate of modern mathematicians. With Archimedes, that demonstration would have ranked heuristics. In the future, Philosophy will object the diagram: an area infinite by extension and finite by size… {2c} This corresponds to a paradox in space which appeared about three centuries later{3d, 5b} and made Thomas Hobbes{3e, 6} say: “To understand this for sense, it does not require that a man should be a geometrician or logician, but that he should be mad.”

12

2 Successful experiments on infinite summation

1 4

1 2

1

1 2

1 4

1

Fig. 2.1. Oresmus evaluates a geometric series “geometrically”.

Oresme’s argument for (1) is “pseudo-geometric”. Its purpose was to answer a problem in pseudo-physics devised at Oxford. To this end, Oresme made the Tower of Babel in Fig. 2.1 stand for a velocity-time-diagram of a piecewise uniform motion, with the straight border line a scale of unit time intervals together with areas that measure the distance covered on the respective interval. The problem reads: Let the velocity be bisected from time to time unit, how does the initial distance “1” compare to all the distance covered thereafter? They are equal. This Oxford problem features time increasing arithmetically and velocity decreasing geometrically. There is a much more sophisticated one on a piecewise uniform motion where time and velocity change places, with time decreasing geometrically and velocity increasing arithmetically. Swineshead (or Suiseth){2e} who had set the distance problem gave a verbal solution{2f, 4c}. Oresme made it “obvious”{3g, 4d}. By appropriate subdivision of the graph’s area which we would call changing the order of integration{2d, 3f, 4b}, he provided a demonstration for the “theorem” 1 ∞ Σk=1 k(2)

k

= 2.

(2)

It shows the geometric series (1) coping with unbounded weights attached to its terms. In our days, (2) is easily accomplished through calculus (3.2.4). However, our blackboards still display Oresme’s hand when it comes to prove the divergence of the harmonic series.{2g, 4d, 7} He discovered it, thus providing an instructive eye opener to those who care little about series convergence if only the terms tend to zero. (In terms of satire: “A series converges if the first term decreases.”) − This series is named according to the geometric one in that each term from the second is the harmonic mean of its neighbors (cf. Appx. A-6). {1} {2} {3} {4}

CAJORI 126. JUSCHKEWITSCH: {2a} 346. {2b} 394 f. {2c} 409. {2d} 408 bottom, 403. {2e} 403. {2f} 403 f. {2g} 410 f. KÖRLE: {3a} 25. {3b} 26. {3c} 28 f. {3d} 41, 143 f. {3e} 41. {3f} 133–135. {3g} 134. SONAR: {4a} 131–133. {4b} 143–145. {4c} 143 middle. {4d} 144 f (Abbn. 4.3.12/13).

2.2 Freestyle: Cavalieri summing indivisibles {5} {6} {7}

2.2

13

SOURCE BOOK … A, (1200–1800) / STRUIK: {5a} 232 top. {5b} 227–230, 230 (Scholium).

STILLWELL 103. BOYER 293.

Freestyle: Cavalieri summing indivisibles

In quite different ways, Archimedes and Oresmus were adding up an “infinite number” of sequential terms. Still, any one was a regular quantity. On the threshold of calculus, even aggregates structured in the sense of Democritus were revived. Recall the ambiguous nature of their elements: Should the indivisibles of a plane figure, for instance, be bare fish-bones or a tiny bit more? Democritus could not give an answer.{1} Archimedes would not. The Italian Cavalieri (cf. 1.2), after having changed his mind at times, gave a pragmatic answer as to the nature of his indivisibles: HOW they work says WHO they are. Period! And he made them work strikingly well, because – and that was quite a novelty – Cavalieri’s approach was systematic. Compared to Archimedes who squared the segments of a single curve, Cavalieri engaged in squaring the so-called higher parabolas when, in his very way, he figured the areas under the graphs of y = x2, x3, x4, ... , x9, to be continued in principle. Galilei, master and friend of Cavalieri, was so impressed by what had been achieved through indivisibles that he asked his disciple to work out a theory. The book Cavalieri wrote did not find many readers{2a}, yet there was steady influence upon analysts including the fathers of calculus. For the purpose of comparison, we will revert to Archimedes. Sharing the success, they both could not differ more than by the access to the problem they had in common: squaring the proper parabola. In 1.3, we sketched the laborious way Archimedes had to go. Cavalieri invites us to follow his short cuts. Here are three. * * * Demonstration 1. In order to represent the area A under the graph of y = x2, 0 ≤ x ≤ 1, Cavalieri constructs an “infinite series” termed by Cavalieri’s indivisibles. He imagines the interval to be subdivided into infinitely many infinitely short line segments with the symbolic unit length α. Every abscissa x is considered some multiple nα which, being assigned the ordinate x2 = (nα)2, contributes to A by the planar indivisible α(nα)2. However, how should a number A evaluate the symbol ∞ Σn=1 α(nα)2

(1)

which does not make sense by itself? Cavalieri puts the prospective value A in proportion to the area S of the unit square with the latter being expressed according to (1), that is as the ∞

∞

product of the base length 1 = Σn=1 α and the ordinate 12 = ( Σn=1 α)2. That, as shows 2 −1 A N(N+1)(2N+1) N N N 3 2 2 , S = α Σn=1n [αΣn=11 · α (Σn=11) ] = 6N3

1

1

makes the dubious α cancel out and results in A = 3 S = 3 , for “N infinitely large”.{3} * * *

14

2 Successful experiments on infinite summation

The indivisibles employed in the sample above are not the ones Cavalieri is known for. To prepare for the following, we need to take a closer look on these. Archimedes, for heuristics, had imagined areas and solids respectively composed by parallel linear and planar segments. He did not believe in the indivisibles of Democritus, nor did Cavalieri though he used the word as a trade mark. His indivisibles{4a}, looking like the cuts of Archimedes, came into being as elements of collections, the “entities”, which he did not want confused with areas and volumes: Those were rather intended to be objects of an extended algebra, sharing properties with sums, infinite series and integrals.{2b} For instance, the bottom triangle of Fig. 2.2 constitutes an entity of indivisibles x denoted Σx within this context. Meaning “all” and applying to solids as well, the symbol is used in the sense of an integral and appears in literature wherever Cavalieri’s wordy exposition is to be concisely reproduced. We see Cavalieri walk between infinite summation and integration, which is typical for those days that prepared the Calculus of Leibniz who likewise collected “all ordinates”, termed “omnes”. * * * Demonstration 2 (in using the pyramid formula). Cavalieri considers the said area A as the entity of ordinates x2, 0 ≤ x ≤ 1. By Fig. 2.2, these ordinates are interpreted as the areas of squares that fill a pyramid and thus form another type of entity.{4b, 5a} Though in different currencies, the area A counts as much as the pyramid (cf. 1.2):

Σ x2 =

1 2 a · 3

a.

(In Demonstration 1, indivisibles owned the traditional quality of areal stripes under the curve, here they form the pyramid’s planar layers parallel to its quadratic base and at distance x from its top.) – – a

a x

x

a Fig. 2.2. Cavalieri: The parabola’s area as a pyramid’s volume.

* * * The next Demonstration will obey the proper nature of Cavalieri’s principle{4c, 5b}. For its introduction, let us consider the trivial case y = x1. In Fig. 2.3a, the square is identified with the entity Σa = Σ(u + υ) which splits into the entities Σu, Συ , the lower and the upper triangle. Plainly, the diagonal bisects the square, by Euclidean congruence. However, Cavalieri aims at Σu = Συ for the sake of Σa1 = 2 Σx1. This is performed by applying his principle as follows, referring to Fig. 2.3b: The triangles, after being turned about A and B by right angles, get crossed by lines parallel to AB each of which cuts out of them a pair of equal seg-

2.2 Freestyle: Cavalieri summing indivisibles

15

ments x. Together with their positions, they qualify for “equal indivisibles”, and so do the two within the right hand square after returning the triangles to their former positions (see Appx. B-1 for further comment). x x

a

x

u

x A

Fig. 2.3a. The entities Σu, Συ.

B

a 1 Fig. 2.3b. ∫ 0 x1 dx = 2 a2 through Cavalieri’s principle.

Demonstration 3 (in proving the pyramid formula).{4c, 5c} According to the pattern ahead, Cavalieri sets up the entity “cube” in terms of Σa2 = Σ(u + υ)2 = Σu2 + 2 Σuυ + Συ2. Here, Σu2 = Συ2 ≕ Σx2 is asserted through applying Cavalieri’s principle to two pyramids with equal bases and heights (cf. Fig. 2.2). When further comparing pyramids this way{5c}, he gets 2 Σuυ = ½ (Σa2 − Σx2) with x in place of u and υ. This makes eliminate 2 Σuυ above and arrive at Σa2 = a

3 Σx2 which yields the volume of the pyramid in Fig. 2.2, now reading ∫ 0 x2 dx = ⅓ a3. (Note

that Democritus had anticipated Cavalieri’s principle.{5d}) – –

Cavalieri’s message was that all parabolas y = xp could be likewise squared. As the exponents increase, the proofs no longer allow for presentation. On this occasion, John Wallis also had to say “and so on”. It was Blaise Pascal who found an approach that convincingly mastered the integration of all the natural powers.{5e} He safely walked the route to infinity, leaving no questions other than those that could only be answered two centuries later. Pascal would do without infinite series. Pierre Fermat put confidence in them when attacking noninteger exponents, and again the geometric series did the job. Though less convincing than Pascal’s proof, Fermat’s idea extended Cavalieri’s formula

Σ xp =

1 1 p+1 p p+1 Σ a (= p+1 a )

to all rational powers xr, r > −1.{5f} The venture Cavalieri took was joined by another disciple of Galilei’s: Evangelista Torricelli. They all were conscious of how far they deviated from Archimedes, yet they all were anxious for discovery. As long as analysis had to wait for its legal foundation nothing could be illegal. Says Cavalieri: “Indivisibles are easier to handle than exhaustions.”{9} Sure.“Rigor is the concern of philosophy rather than of mathematics.”{9, 10} No comment. {1} {2} {3} {4} {5}

VAN DER WAERDEN [1] 138; [2] 228. ANDERSON: {2a} 18. {2b} 18 ff. REIFF 4 f (misprint p.5 middle). EDWARDS: {4a} 104. {4b} 106 f. {4c} 107 f. KÖRLE: {5a} 145 f. {5b} 39, 141. {5c} 146. {5d} 147 above, 101 f. {5e} 147–150. {5f} 151 f.

16

2 Successful experiments on infinite summation

{6} {7} {8} {9} {10}

VAN MAANEN 70 top. KÖRLE: {7a} 39, Abb. I.8a. {7b} 39, Abb. I.8b. SOURCE BOOK/ STRUIK: {8a} 215, proof part 1 (beware misprints). {8b} 218, Fig. 3. HEUSER [2] 650. KLINE 383. THIELE 35.

2.3

Leibniz’ telescopes

The first one to make Leibniz get in closer touch with mathematics was the Dutchman Christiaan Huygens, a celebrity by that time and one of the very few that had won Newton’s appreciation. He might have put Leibniz to the test when asking him to “sum” the reciprocals of the triangular numbers{1a} that count the grid points covered by subsequent triangles according to Fig. 2.4. They are the simplest type of figurate numbers{1b} (further formed with regular polygons and polyhedrons), being in vogue with the Pythagoreans.

Fig. 2.4. Triangular numbers 1, 1+2 = 3, 3+3 = 6, 6+4 = 10, which is

k(k+1) 2 ,

k = 1, … ,4.

Huygens’ invitation to Leibniz presupposes that

Σk∞=1 k(k1+1) =

1 1 1 1 + + + + 2 6 12 20

…

(1)

is sparse enough a subseries of the divergent harmonic series in order to be convergent. What makes the difference? The difference. When playfully having started the scheme{1c, 2a}

∆

1 = 1 k 1 1 = 2 k

1 2

1

1 3

∆2 k =

1 3 1 6

1 4 1 12

1 12

1 5 1 20

1 30

1 … 6 1 … 30

(2)

1 …, 60

Leibniz noticed the terms of (1) to appear in the second row. At second sight he saw the 1 1 1 fractions k(k +1) split into their partial unit fractions k , k+1. To him, it was fascinating to watch the terms then superpose like the segments of a telescope when sliding together (Fig. 2.5). So he had no scruples to see the series readily shrink down to its first term: 1 1 1 1 1 1 1 Σk∞=1 k(k1+1) = Σ∞ k =1(k − k+1) = (1 − 2 ) + ( 2 − 3 ) + ( 3 − 4 ) + … 1

1

1

= 1 + (− 2 + 2 ) + (− 3 +

1 3

) + 0 + 0 + … = 1.

2.3 Leibniz’ telescopes

17

1 +

4 – 1

+

9 – 4

+

Fig. 2.5. The telescope effect on 1 + 3 + 5 + 7.

16 – 9 = 16

Actually, it’s a matter of two different series, i.e. series that differ by their terms. The construct above is not self-evident as is suggested by the operations on 5.2.1(1). We better telescope the partial sums and write 1 Σnk =1 ( 1k − k+1 )=

1 1 − n+1 → 1 as n → ∞ .

(3)

(For evaluation or comparison of series, finite or infinite, it may pay off to convert them into telescopes. Cf. 3.2.1(7).) Leibniz’ evaluation was attributed to Huygens, it had been anticipated by the Italian Pietro Mengoli.{3a} Leibniz wouldn’t have been Leibniz, if he had been content with this application of the idea. Just as the terms in the second row of (2) add up to the leading term 1 of the first row, so the third one results in the value ½ by which the second starts, and so forth. We may look at system (2) from another point of view, from south-east so to speak, and take the sums of adjacent terms: 1 1 + 2 2

1 3

= 1,

1

1

1

1

+6=6+3=2,

1 1 + 12 12

1

=6,

1 1 + 12 4

1

=3, ….

It reminds of the way Pascal’s triangle (within Fig. 3.5) is generated successively, by 3.2.3(3). If the matrix in (2) is given a quarter turn, clockwise about its vertex 1, an analogue to Pascal’s triangle appears with the harmonic sequence on its flanks. Leibniz calls it the harmonic triangle.{1d, 2b, 4} * * * Partial fractions and the telescope effect also yield ∞

1

∞

2

2 Σk = 2 k2−1 = Σk = 2 (k+1)(k−1) =

1 1 Σk∞= 2 (k−1 − k+1)

1

1

= 1 + 2. ∞

(4)

By its appearance, the series (4) just slightly differs from Σk =11/k2 (for convergence, note that (k +1)–2 < k–1 – (k +1)–1). After having evaluated “his” series 3.2.1(1), Leibniz would contend able“to sum any series”. He, the Swiss or rather Dutch Bernoulli brothers Jacob (Jacque/James) and Johann (John) as well as others like Mengoli{3b} had been trying hard to evaluate the latter series. Only Euler found the solution − and what kind of a value: π2/6 (3.2.6(1)). {1} {2} {3} {4}

EDWARDS: {1a} 236. {1b} 111. {1c} 237 f. {1d} 237. SONAR: {2a} 407. {2b} 406. BOYER: {3a} 406. {3b} 486 f. KÖRLE 169, Fig. 6b (the dotting is nonsense).

3

Series of functions: the prototypes

3.1

Preliminary: Aspects

3.1.1

Series versus sequences

An IQ test might ask to continue the sequence 1, 1.5, 1.833... . Indeed, one rather continues ∞

the series that is associated with it. Recall: Any series Σk=0 uk generates the sequence sn =

Σnk=0uk of its partial sums, the terms of any sequence s0, s1, … are the partial sums of the ∞

series Σk=0(sk – sk–1), where s–1 = 0. (The geometric series and its followers account for the habit to make any series and sequences start with the zero term if not suggested otherwise.) Traditionally, “infinite series” comprises of diversity. (“Series” commonly stands for infinite ones.) It makes no pseudonym if we include the ancients’ substitute by the name of exhaustion, a sequential approach. We saw Eudoxus (1.2(3)) and Archimedes (1.3(3)) virtually construct infinite series. With the Hindus and the Oxford school they became outspoken. Centuries later, Newton made them his own very tool, a much formal but most effective one. Those who then applied it did not question whether series deserve to be treated like sums. Euler and Fourier would not worry about the difference when they opened new horizons to analysis. When, in the end, doubts were arising the Greek approach through sequences came in vogue again. One may hear people say that infinite series and sequences are all the same. They are, in principle. They acted and still act different roles in the same play. When “Greek” was out of date, the language of series was in. They raised the problems, made the fund of analysis apply on the spot. Sequences became the domain of theory. * * * ∞

Modern calculus has its definitions, their use is eased by habits. With the series symbol Σk=0 uk it is just the sequence (u0, u1, u2, … ) of terms that matters, properly said a well defined mapping k ↦ uk from ℕ0 into numbers (equal ones in a sequence must be well distinguished). The sigma is used in various ways. It communicates the sums sn, something concrete to work with. When it occurs under an operation or within a relation, then convergence is provided or assumed to hold and renders the whole entitled to represent a number, the limit of the partial sums. On the other hand, if one asks “Does the series Σuk converge?”, then the symbol reduces to form, intended to ∞

address the partial sums of any series Σk=k uk , k0 ≥ 0. 0

20

3 Series of functions: the prototypes

“Formal” often bears a pejorative connotation. There is hardly form without any substance. In 3.2.6, we will watch Euler operate formally on infinite series in the course of his “algebraical analysis”, most effectively though. As long as we are conscious of what we are doing, we may do anything to a series, for what doesn’t work formally won’t work substantially.

3.1.2

Series: evaluation and expansion

Decreasing geometric progression apparently insinuated infinite series (see 1.3, 2.1, 3.2.1{1a}). Our kids come into contact with them when they learn about periodic decimal frac∞

tions, a funny sort of aggregates Σk=1 ck qk with q = 1/10, ck = 0, ... ,9 (cf. Appx. A-6). Divisions like 1 : 10, 1 : 8 break off whereas 1 : 9 does not: a “long division” turns out that yields an infinite series of digits. By childish experience an “infinite series” is born in the name of analysis: 1 9

= .111 … = .1 + .01 + .001 + … =

k

1 Σ∞ 1·( ) , k=1 10

(1) 1

a geometric series. For instance and for exercise, look for a corresponding “expansion” of 7! An equation does, its genesis may have two sides. We just started from the left of (1). Let be 1

given the infinite decimal fraction d = .111 … , how does it reproduce the proper fraction 9? The teacher will resort to this: 10 d =

1.111 …

– d = – .111 … 9d =

1,

(2) 1

hence d = 9 .

Hence, equation (1) may be considered either a proper fraction being expanded into an infinite decimal fraction or this infinite series being evaluated by the “closed” form of a proper fraction. Both points of view are assumed in a theory of infinite series. The computer (2) correctly reflects analysis. It might turn out to be pitfall. To a teacher, when challenged by a youngster who makes it operate on d = .99... and wants him to explain the output: .99 … = 1.

(3)

Is it higher mathematics in the sense of nonsense, the geometric series a fake? It is one of the tricks the infinite plays on the human mind. (In fact, (2) is advanced calculus; see 4.3 for clearance.) * * * We are trained to think in terms of variables: The geometric series is a series of functions. ∞

Values of Σk=0 xk like 4/3 and 2 in case of x = ¼ (from 1.3(3)) and x = ½ (from 2.1(1)),

3.1 Preliminary: Aspects

21

respectively, were found to specialize from the same algebraic expression, the ratio 1/(1 – x). As the calculation (2) looks persuasive one might be tempted to regard the procedure s(x) = 1 + x + x2 + ... x + x2 + ...

x s(x) =

(4)

(1 – x) s(x) = 1 1

result in s(x) = 1 – x as long as x ≠ 1 (?!). We know that the geometric series cannot converge in case of | x | ≥ 1 since then its terms do not form a null sequence. It took quite some time in history until such concern developed with infinite series. The pattern (4) leads up to Archimedes when being applied, instead, to the partial sums

Σnk=0 xk. Their closed form in 1.3(3) – to him it was the means of precisely controlled approach, to us its limit readily yields the evaluation ∞ k Σk=0 x

1

= 1 – x , –1 < x < 1.

(5)

From the other point of view, (5) is interpreted to be a power series expansion of 1/(1 – x) ∞

when restricted to the domain (– 1, 1). Asking for any power series Σk=0 ak xk to take the part of the geometric series in (5) we will learn: there is no one else! But how does (5) result when started the other way around, from the right that is? The Scotchman James Gregory{2a} and Sir Isaac Newton{2b} transferred long division, like at (1), from integers to polynomials when they wrote

1 : 1  x 1  x  x 2 } x 1____ x x  x2 ____ x2 }

 Fig. 3.1. Long Division.

This, when being given the form 1 = (1 – x) (1 + x + x2 + ... ), reflects (4). Above we saw a special power series generate a function which, in turn, regenerated that ∞

series. For series Σk=0 ak xk and the general power series (see below) we will ask: Given a power series, does it converge so as to generate a function? Given a function, does it generate a power series that converges to the values of the function? * * * Archimedes evaluated an area through virtually evaluating an infinite series, a power series. He also practiced non-serial partition (circle, solid sphere{3a}) and was followed in this by for

22

3 Series of functions: the prototypes

example Pappus and Kepler{3a} – it is the approach akin to integration. Infinite series were handled in a direct way by Oresmus (2.1). Then, variable terms were in the air: A Hindu (3.2.1) and, independently, Isaac Newton had the idea to employ power series expansion. Complex terms soon became standard; since Euler and Cauchy, the “complex number plane” will be the place where power series feel at home. Power series usually mean series termed ak(z − z0)k where k = 0,1, ... . Negative exponents k indicate poles; so, it comes to inquire into infinite series of powers that include negative ones. They prove indispensable to analyzing “singularities” of functions (3.4.1). In principle, the denotation “power series” is open to this connotation.{4} Early in the 18th century, quite another type of function series made its appearance (3.3). According as ... , z−2, z−1, z0, z1, z2, ... form power series, the trigonometric series are based on the scale cos(0 x); cos(1x), sin(1x); cos(2x), sin(2x); ... . (As is commonly accepted, write cos kx for cos(kx) and the like.) Each of such “bases” shares some property with, say, the canonical base of the Euclidean vector space ℝn. We exclusively deal with those two types of function series because, to a large extent, ∞

they account for the course that analysis took until about 1900. A series Σk=0 ak (z − z0)k, convergent for some z ≠ z0, has derivatives of any order at z0, whereas the values of a trigonometric series may show all sorts of strange behavior. Above, we used power series to illustrate evaluation and expansion. Both aspects are offered by trigonometric series in a much more intricate manner. The caprices of these series proved to be quite a virtue: They serve as candidates to represent irregular real functions. Moreover and in particular, their intrinsic problems became incentive for a lot of today’s mathematics.{1b} Power and trigonometric series have contributed in specific ways to complex and to real analysis. Where evaluation is concerned, there is another aspect to be cleared, a most vital one. Series may be formed arbitrarily and they would originate by nature. People had an intuitive idea about calling them convergent, convergent to a value. Analysis was not prepared to answer the crucial question in general: What is in any case, a value? A number. A number? What is in any case, a number?? In principle, yet in principle only, the question ought to be answered before one could justly have a power or trigonometric series generate a function. (For how the deficiency was eventually made up, read 4.3, up to the end.) This aspect is being ignored in the threefold treatise {5a} on infinite series: cf. {5b} for convergence and divergence. (Otherwise, these courses provide all the standard reference.) {1} {2} {3} {4} {5}

SONAR: {1a} 347. {1b} 554 f. {2a} BOYER 423. {2b} EDWARDS 186. KÖRLE: {3a} 115 ff. {3b} 132 f. PÓLYA; LATTA 199 top. NEEDHAM [2] 464 bottom. FICHTENHOLZ; SILVERMAN: {5a} [1, 2, 3]. {5b} [1] 2, DEFINITION.

3.2 The power of power series

23

3.2

The power of power series

3.2.1

Nīlakantha and the arcustangent series

It might be worthwhile to expound a sample of infinite series from outside the European tradition, all the more as it anticipates, by almost two centuries, some of the calculus developed by Newton, Leibniz and their contemporaries. Gottfried Wilhelm Leibniz, a self-made newcomer, took much pride in having discovered the Leibniz series that represents π in terms of natural numbers{1a, 2a} and is therefore called the arithmetical quadrature of the circle {3a}: π 4

1

1

= 1 −3 + 5 − + …,

(1)

where the ratio π : 4 tells us how a circle and its circumscribed square compare by their areas as well as their circumferences. Leibniz reads (1) from left to right. The other way means evaluation of series. Who calls it a “closed evaluation” must be conscious of what in fact happens: The value of the series is a limit, it is expressed by another limit ; cf. 3.2.6.) This famous accomplishment, gained by an essential{3b} of his Calculus, opened Leibniz’ career as a mathematician. Shortly after, in 1673{2a}, Leibniz knew to formally embed his result in x3 x5 arctan x = x − 3 + 5 − + … , 0 ≤ x < 1,

(2)

by x = 1. Well, this power series converges on all of [0,1] {3c} as is due to the well known Leibniz test on alternating series, however it is Abel’s limit theorem (Thm. 4.2) that extends (2) to x = 1! Leibniz would not wait for Abel: He asserts (2) to infer (1) on the grounds that “what holds up to the limit is true for the limit” (cf. 5.2.1; our freshmen would make the point 1/n > 0 = lim 1/n ). As early as in 1671{1a,b, 4a}, James Gregory had already detected (2) and so the arcustangent series came to bear his name. Through priority. (But we ought to be cautious as for the chronology of discoveries in mathematics.) Leibniz traced back the expansion of arctan x to a geometric series when he applied his suggestive formula dy/dx = (dx/dy)−1 to the function x = tan y : 1 d arctan x d tan y −1 2 4 = ( dx dy ) = 1+ x 2 = 1 − x + x − + … , | x | < 1,

(3)

which “only” needs to be integrated term by term in order to yield (2). In history, systematic differentiation falls behind integration by about two millenniums, so Gregory started out from what he might have learned in Italy{1b}, namely x

∫ 0 1 +dtt 2 = arctan x,

(4)

and then expanded the integrand. Yet, was Gregory really the first among the fathers of the arctan series? * * *

24

3 Series of functions: the prototypes

Once upon a time, in southern India during 1501/02{1c, 2b} to be precise, a renowned scholar by the name Nīlakantha wrote his “Scientific Collection” that includes the present topic.{5a} In the times of Newton and Leibniz, “Explanatory Comments”{2b} anonymously appeared which was considered to report fairly well{2c} the ideas ascribed to the author (or some predecessor{5b}). There was need for comment, since the style of the Collection rather differs from Archimedes’ writings: Nīlakantha conveys the results alone, and he does so through Sanskrit verses, like a means of consecration. In our notation, he asserts that, when proceeding by tan φ –

tan3φ tan5φ 3 + 5 −+ … ,

(5)

“you will end up with φ itself ”{2d} (supposed φ ≥ 0 to be restricted “sufficiently”). This, with φ = π/4, makes one arrive at (1). We will refer to Nīlakantha’s proof as being presented by Youshkevitch{2e} (cf. {4b, 5c}). As (3) and (4) show, Gregory and Leibniz walk back and forth on the bridge between the said function and the geometric series. This is precisely what Nīlakantha did: differentiate and integrate. * * * For differentiation, look at Fig. 3.2 where some φ ∈(0, π/2) is assumed fixed. As Δφ → 0, we see Pφ+Δφ approach Pφ along the unit circle and Tφ+Δφ slide down the tangent to Tφ . With the line segments Mφ,ΔφTφ , Nφ,ΔφPφ (which read MTφ , NPφ in Fig. 3.2) perpendicular to the line OPφ+Δφ , there is ( sin ∡OTφ+ΔφP0 = ) |Mφ,ΔφTφ| : Δ tan φ = 1 : |OTφ+Δφ|, ( sin Δφ = ) |Mφ,ΔφTφ| : |OTφ| = |Nφ,ΔφPφ| : 1. TM + 'M

' tan M M

PM + 'M

TM

N PM

'M O

M

tan M P0

1

Fig. 3.2. Nīlakantha “differentiates” the tangent function.{2f, 4c, 5d}

Observing |OTφ+Δφ| ≈ |OTφ| , |N φ,ΔφPφ| ≈ Δφ for “small Δφ”, the two proportions yield Δ tan φ

|Mφ,ΔφTφ| ≈ | OT | ≈ |OTφ| Δφ and thus φ

Δ tan φ 2 2 Δ φ ≈ |OTφ| = 1 + tan φ ;

3.2 The power of power series

25

in terms of x = tan φ, the latter reads Δ arctan x 1 ≈ 1 + x2 Δx

(6)

in accord with (3) and (4). With a fixed value of x = tan φ ≥ 0 to be specified by later need, Nīlakantha would start (cf. Fig. 2.5) from N

φ = arctan x = Σn=1[arctan xn,N – arctan xn–1,N] , where xν,N =

ν N

x , ν = 0, … , N.

(7)

Applying (6) to xn ≡ xn,N , Δxn = xn − xn−1 , he infers φ ≈

n ΣNn=1 1 Δx + xn2

≡

x N

ΣNn=1 1 + 1xn,N 2

(8)

with the right side supposed to approach φ indefinitely close as N → ∞. At this point, it’s time for geometric expansion which then was given the notation 1 1 1 2 1 1−q = 1 + q 1−q = 1 + q (1 + q 1−q ) = 1 + q + q 1−q = … . This applies to the terms in (8), provided that 0 ≤ xn < 1, and makes (8) assume the form φ ≈

x N

p n 2p ΣNn=1 Σ∞ ( −1) ( N x) p=0

=

p 2p+1 Σ∞ ( −1) x [( N1 ) 2p+1 ΣNn=1 n2p]. p=0

To arrive at φ by N → ∞, Nīlakantha has to invert the order of limits: ∞

φ = Σp=0 (–1)p x

2p+1

[lim N→ ∞ ( N1 ) 2p+1 ΣNn=1 n2p].

(9)

As then was taken for granted or ascertained some way {2g, 4d} – and eventually proved by the Frenchman Blaise Pascal{5e, 6a} –, the limits in (9) exist by the values 1/(2p +1). This would verify Nīlakantha’s claim according to the standards of his time. His concern was π. At (1), the mode of convergence is awfully slow: More than a million terms are required for 6 decimals assured{7a}, and to even catch up with the accuracy Archimedes had attained, it allegedly needs about 100.000 terms{7b}. Nīlakantha successfully tried out other approaches, particularly operating on (1){2h}. Leibniz will comment on his discovery of (1) saying: “Never before, anybody has succeeded in squaring the circle arithmetically.” {6b, 8} Never say never. * * * Archimedes had found the way to approach the mysterious π through rationals as close as one ever wants.{7a} Nīlakantha succeeded through unit fractions. He, like Leibniz, might say to have found π itself. They brought together quantities which do not share quality except for aesthetical regularity. The discovery evokes Archimedes breaking the dogma on the crooked and the even (see 1.3).

26

3 Series of functions: the prototypes

For good reason, Leibniz felt enthusiastic about his series. So did I when meeting with it. Wasn’t it unbelievable? How could geometry and arithmetic thus join? Presumably, the miracle was given adequate Sanskrit expression. * * * The great Nīlakantha was not alone in India when he cultivated an esoteric artifice of communication: laconic style, little, if any, proof and poetry at the expense of information, let alone didactics which then was reduced to learning verses by heart.{2i, 8} Where transparency is concerned, mathematics in India lacks that culture of proof which was the exorbitant trend of Greek mathematics{9} from the outset, culminating in the work of Eudoxus and Archimedes (1.2). This would have had an end had the Greek tradition not been taken care of and continued by Arabian scholars{2j, 6c} who closed the gap of more than a millennium by providing Western access to Greek literature. Within this period, Hindu mathematics widely developed on its own and without the drive for mission, whereas the Arabs imported, adopted and exported any Hindu knowledge like astronomy, arithmetic, algebra, not to forget the Indian “Arabic numbers”{6d}. Their activity opened a richly assorted bazaar in Europe. {1} {2}

{3} {4} {5} {6} {7} {8} {9}

3.2.2

BOYER: {1a} 443. {1b} 422. {1c} 422 footn.10. JUSCHKEWITSCH (YOUSHKEVITCH): {2a} 173. {2b} 167. {2c} 169. {2d} 168. {2e} 169 f. {2f} 170. {2g} 170 bottom, (13). {2h} 168 f, 171 ff. {2i} 92 f. {2j} 175 ff. EDWARDS: {3a} 245. {3b} 246–248 (Fig.5 is erroneous). {3c} 248 bottom line?! SONAR: {4a} 346, 347 bottom. {4b} 347–350. {4c} 348 f. {4d} 349. RAJAGOPAL: {5a} 202. {5b} 203. {5c} 203–205. {5d} 204. {5e} 205 (Lemma 3). KÖRLE: {6a} 150. {6b} 59. {6c} 21 f. {6d} 22. {6e} 115–119. {7a} KNOPP [1] 261; [2] no.145, 4. {7b} KLINE 439. CAJORI 83. SCRIBA 115.

The hyperbolic logarithm and its series

The main event to bring forth the culture of calculation must be considered the invention of the Zero. Its career started two millenniums ago and culminated in the zeros and ones of today’s number notation. In between, there was another happening: the invention of logarithms.{1a} Michael Stifel would figure 32·28 = 25·27 = 25+7 = 4046, changing multiplication into addition. (He coined the word exponent.) No sooner than by the end of the 16th century this pattern gave rise to benefit in general from what in case of l(2n) = n reads l(u·υ) = l(u) + l(υ), l(1) = 0.

(1)

Independently, a Swiss and a Scot gave answers: Jost Bürgi{1b, 2a} and John Napier{1c, 2b}. Without exponential notation they translated “numerals” (numeri) N into “logarithms” L so as to reflect the relation N = bL with some positive “base” b ≠ 1. Both made their choice on b and developed a table{3a} of discrete values N, L. Napier did more. He conceived a continuous version of the pairing (N, L) {2c, 3b}, in a peculiar fashion, turning out{3c} however to be our

3.2 The power of power series

27

system of natural logarithms (3.2.5)! It surely was under a lucky star, the day when the London professor Henry Briggs paid a visit to Napier, the Scottish baron.{1d} They, within the very day, decided to make the decimal base of numbers a base of logarithms. Decimal or Briggian logarithms caused enthusiasm with people like Johannes Kepler. In the words of F.Cajori{1a}, the invention of logarithms “by shortening the labors doubled the life of the astronomer”. * * * The first lasting access to natural logarithms dated no later than 1647{2d, 3d} when Grégoire de Saint-Vincent and his disciple Antonio de Sarasa happened to make great a discovery on the classical hyperbola of Apollonius, the equilateral or unit one. Here is, in view of Fig. 3.3, what they found out (cf. {3d}): Let A[a, b], 0 < a ≤ b, and the like denote the area under the curve over the interval; there holds A[a, b] = A[q a, q b] ,{2e, 3d} with whatever q > 0 (thus allowing the intervals to overlap). Now, take a = 1 and let L(x) ≔ A[1, x], x ≥ 1; then, with u, υ ≥ 1,{2d, 3e} L(u·υ) = A[1, u·υ ] = A[1, u] + A[u, u·υ ] = A[1, u] + A[1, υ ] = L(u) + L(υ), L(1) = A[1, 1] = 0

(1*) x 1

according to (1). (For 0 < x < 1 see {2d}. There, L(x) ≔ – A[x, 1], which yields L(x) ≔ ∫ 1 t dt, x > 0, in general.) This accounts for the function L to be named the hyperbolic logarithm. Pietro Mengoli{4a}, “unappreciated”{4a} at his time, had named it “natural logarithm”{4b}... Euler will take up the word, for good reason (3.2.5). A century later, we will see Euler interpret the hyperbolic logarithm his way, i.e. pertaining to his mysterious number “e” (3.2.5). He is the one who would definitely base the values L(x) on a “base”, namely with reference to x = eL(x). Since it soon came into common use, we will henceforth call L(x) the natural logarithm{3f} of x and write ln x, where x > 0. 1 x

1 0

A 0

a1

b 3a

A

3b

x

Fig. 3.3. Saint-Vincent and the hyperbola of Apollonius.

28

3 Series of functions: the prototypes

1 x

1 0

A 0

q0

q1

A

A

q2

q3

x

Fig. 3.4. Geometric versus arithmetic progression.

By (1*), there is L(uk) = k L(u). When u > 1 we get geometric growth of numerals corresponding to arithmetic growth of logarithms. To watch hyperbolic logarithms share this, see Fig. 3.4. With q > 1, k = 1, 2, ... , the intervals [qk–1, qk] increase geometrically by their lengths qk – qk–1 = (1 – 1/q) qk, whereas the areas on them keep constant and thus trivially proceed arithmetically (see Appx. A-6). Furthermore, when joining the intervals, we note the quasi exponential progression of qk – q0 = (1 – 1/qk) qk to go with the arithmetic progression of the areas k A on [1, qk] = [1, q] ∪ [q, q2] ∪ ... ∪ [qk–1, qk ]. Another comment on Fig. 3.4: Look for the value of q that makes A = 1 – – well, this is Euler’s q = e (Def. 3.1) in Fig. 3.7. * * * Euler will give the power series expansion of the natural logarithm in the context of 3.2.5. Before, it had been accomplished on terms of the original hyperbolic one. Nicolaus Mercator{4a, 5a}, inspired by Cavalieri’s indivisibles{5b}, may have claimed to be the first to have acquired what, in our diction, writes x2

x3

x4

ln (1 + x) = x − 2 + 3 − 4 + − … .{2f, 5c}

(2)

This was pointed out valid for | x | < 1 by John Wallis the referee who also gave appropriate proof.{1e, 2g} Another follower of Cavalieri’s, Mengoli{4a} (cf. 2.3) who later converted to infinite series, had adopted from (2) the series representation of ln 2 which Abel will verify by his limit theorem (Thm. 4.2). Mercator’s series became anticipated otherwise{4b}; in particular, Newton{2h} obtained x

∫ 0 1 1+ t dt

=

x

∞ ∫ 0 Σk=0 (–t)k dt

=

x

Σ∞ (–1)k ∫ 0 t k dt k=0

=

k

x Σ∞ (–1)k−1 k , k=1

(3)

afterwards being recognized as Saint-Vincent’s{2i} logarithm. The latter had once been an ingenious speculation in geometry; Wallis and Newton made its sort of dubious descent forget.{5d} The logarithmic series was to excel for its theoretical importance in analysis. In order to become a means of calculation it needed transformation. This new approach meant a great accomplishment{11e} since Napier and Briggs faced painstaking labor to improve upon their tables which had been constructed step by step with all the risk of error. Nowadays, logarithms and their slide rule analogues have become largely obsolete after calculation ends with what it once started: by digits.

3.2 The power of power series {1} {2} {3} {4} {5}

3.2.3

29

CAJORI: {1a} 149. {1b}148, 152. {1c} 149 f. {1d} 150 f. {1e} 156 top, 188. EDWARDS: {2a} 152 f. {2b} 144–148. {2c} 148–151. {2d} 156. {2e} 154 f. {2f} 162 (8); cf. (9), misprinted. {2g} 162 f. {2h} 158. {2i} 159. KÖRLE: {3a} 155 f. {3b} 156 f. {3c} 159 (15). {3d} 153 f. {3e} 158. {3f} 156. BOYER: {4a} 406. {4b} 423. SONAR: {5a} 332–337. {5b} 335, Abb.6.3.23. {5c} 335. {5d} 336 bottom.

Newton and the binomial series

Independently, Isaac Newton and Gottfried Wilhelm Leibniz invented − “detected”, Plato would have said − the calculus of differentiation and integration. According to their motives, they made infinite series play different roles. Mechanics had guided Archimedes to his discoveries in mathematics. When looking for tools in kinematics, Newton found Calculus (we will thus write historical calculus). From the outset he knew how to enhance its efficiency: Once, some function was represented in terms of the variable’s natural powers (3.1.2), he would make Calculus operate on power series the very way it did with polynomials, term by term. For it couldn’t be that risky as long as sound results were thus produced, to be verified not only by physicists. So, in the first place, Newton’s concern was how to make a fractional power be represented by natural ones. For eons, natural powers (a + b)n of binomials had been represented, so to speak expanded, in terms of powered monomials. Probably as early as 1100{1a}, binomial expansions made their appearance in China, along with appropriate patterns for coefficients; a triangle dated 1303 shows today’s arrangement{1b, 2a}. Around 1540 and surely on his own, Michael Stifel did more than this and is likely to have noticed the coefficients’ law of recursive formation{2b, 3a}, namely the students’ way to figure them out. Pascal knew how to make them explicit. His binomial theorem says: (1 + x)n =

Σnk=0 cnk xk,

n = 0,1, … ,

(1)

with the binomial coefficients n(n−1) ... (n−k +1) n n ≕ ( k ) , k = 1, … , n, (2) cn0 ≔ 1 ≕ ( 0 ), n = 0, 1, … , and cnk ≔ 1 · 2 · ... · n the entries of Pascal’s triangle (in Fig. 3.5 below). As for integer powers in general, (1 + x)−1 ∞

is represented by Σk=0 c−1,k xk, | x | < 1, with c−1,k = (−1)k (3.1.2, 3.2.4), a geometric series which Newton could have repeatedly differentiated or, as Cauchy did (3.4.2), multiplied by itself in order to expand the negative powers (1 + x)m, with m = −2 at least. Surprising enough, the geometric series did not play a significant role at that time; Newton came up with it by himself{3b}. Newton must have attacked the powers (1 + x)p/q in general no later than 1665, as might be suggested from the report in {3c} on his unpublished work. They caused disproportional trouble for him. First evidence of the outcome is given in Newton’s two letters from June and October 1676 consigned to be forwarded to Leibniz.{6a} James Gregory is said to have had the result by 1670.{2c} Finally, calculus would provide easy access by Taylor’s series of

30

3 Series of functions: the prototypes

1715{2d} (3.2.4). In 1736, John Colson made that point, followed by Colin MacLaurin five years later.{2e} Newton’s explanatory notes to Leibniz were not quite as explicit as a thoughtful instruction could have been, to say the least. Despite the most courteous and respectful words then accorded to Leibniz, Newton considered or was to consider him a presumptuous novice. * * * The ominous number π which gets involved in so many enterprises also nursed the binomial 1

series when Newton followed John Wallis on his way to the quarter area ∫0 √(1 – x2) dx of the unit circle.{3d} Unfortunately so, for it turned out to be a most laborious approach. Wallis had concerned himself with interpolating the integrals of the powers (1 – x2)0 and (1 – x2)1, Newton focused on the integrands themselves to be interpolated by a power series expansion of (1 – x2)1/2 (thus denoting the square root on this occasion{3e}). Pascal’s formula (2) also covers k = n +1, n + 2, … and could have suggested further interpretation. Had Newton only been aware, or had he taken notice of this representation he sure would have inserted the exponent 1/2 there. Instead, Newton was after an extension of the Pascal triangle by intermediate rows; for the extensive and intricate heuristics see {3f}, commented in Appx. C-1. Pupils are used to constructing Pascal’s scheme row after by row by Stifel’s law cnk + cn, k+1 = cn+1, k+1 (n = 1, 2, … ; k = 0, … , n – 1)

(3)

which Newton the self-made man had to find by himself, according to records{6b}. He then, on speculation, was laboring at intermediate entries cp/2, k such as to fulfill Stifel’s law on their own part, with p/2 = 1/2, 3/2, … : cp/2, k + cp/2, k+1 = cp/2 +1, k+1 (p = 1, 3, 5, … ; k = 0, 1, …) . k = p 2

0 1

= 0 1 2

1



3 2

2

… … …

? 0

… …

? 0

3 8

1

3 0

1 8

0

1

1

2 0

1 2

1

3 2

1

0

1

1

2

1

…

Fig. 3.5. Top of the original Pascal triangle (boldfaced), in the process of being completed by binomial coefficients for (1 + x)p/2, p = 1, 3, 5, … .

Fig. 3.5 reflects the “Tables” on {3g} the source of which is {7}, a facsimile of Newton’s autograph. It gives evidence of the literally painstaking job that preceded the verification

[Σ∞ ( ½ ) xk] k=0 k

2

= 1 – x2,

3.2 The power of power series

31

with Newton anticipating Cauchy’s product. On the grounds of samples and analogy, Newton would assert the result to be valid “in general”{5b} ,“whether that power is integral or (so to speak) fractional, whether positive or negative”{5c}. Except for exponents r where the series shrinks to Pascal’s sum in (1), the expansion (1 + x)r =

∞ r Σk=0 ( k ) xk,

r∈ ℚ ,

(4)

is valid but for | x | < 1. * * * Newton had been the first to fully utilize fractional powers and to put them into effective action. There was a critical voice that questioned the credit given to Newton, asking for how much he may owe to Wallis and perhaps to Stifel and Pascal.{2f} (However, the unpublished work in {6} which {3f} refers to was not publicly known when {2} was written.) His was a pioneer’s work, the more so as Newton had to rediscover quite a lot when starting his career. After all, while Leibniz was reading his learned father’s library, Newton had to herd the pigs. Newton was devoid of scruples when applying calculus operations on infinite series, one reason being that he and his contemporaries were far off the time when convergence could become a solid notion. Being well aware that variables need to be restricted in cases, he tacitly understood them to be assumed sufficiently adapted to the occasion. The master was forging tools to serve his primary interests, so he cared less about explicit justification beyond. Luckily, his constructs kept playing the game mathematically. Nevertheless, Newton the physicist proved to be a brilliant mathematician as well. Through intuition. After what he had experienced, he could not but feel convinced of more than analogy to hold between the finite and the infinite. His was the belief in their thorough harmony. Why distrust his series? Siblings of polynomials as they were, they became first choice among the series of functions (3.2.4, 3.4.1). Certainly more sensitive about convergence was the one who created the word{8a} − and virtually detected calculus no later than Newton and Leibniz: the man from the Highlands, James Gregory{8b}. In the century to follow, Joseph-Louis Lagrange would become a critical analyst. The first one to engage in the convergence of power series was not Cauchy, but the Norwegian Niels Henrik Abel{2g} (4.1). {1} {2} {3} {4} {5} {6} {7} {8}

BOYER: {1a} 227. {1b} 228. {1c} 432 top. COOLIDGE: {2a} 149. {2b} 149 f. {2c} 151 f. {2d} 157. {2e}154 f. {2f} 153. {2g} 156 f. EDWARDS: {3a} 168. {3b} 186. {3c} 180 ff. {3d} 179 bottom. {3e} 180 top. {3f} 179–183. {3g} 181, Table 4, 5. KNOPP [1] 193; [2] 190. SOURCE BOOK ... [1] / STRUIK: {5a} 285–290. {5b} 289. {5c} 286. NEWTON: {6a} 126–134. {6b}128 (footn. 26). {6c} 125. HAIRER; WANNER 24, FIG. 2.4. {8a} CAJORI 228. {8b} SONAR 346.

32

3 Series of functions: the prototypes

3.2.4

Taylor series and expansion; ordinary power series

Among the fathers of the Taylor series, a docile disciple of Newton by the name of Brook Taylor must have fathered it after differentiating a polynomial p(x) = a0 + a1(x − x0) + ... + an(x − x0)n at x = x0 repeatedly: p(k)(x) =

Σnj=k

j (j − 1) … (j − k + 1) aj (x − x0)j−k

=

n−k Σj=0

(j+k)! j j! aj+k (x − x0) (Appx. C-2),

(1)

p(k)(x0) = k! ak + 0 + … + 0 , k = 0, 1, … , n, p(x) =

(k)

Σnk=0 p k!(x0) (x – x0)k.

(2)

The formulas say that the coefficients and thus all values of the polynomial are determined by its values in whatever small a neighborhood (x0 − δ , x0 + δ ) of x0. With this in mind and a function f that allows for all derivatives at x0, Taylor formed the series ∞ Σk=0

f

(k)

(x0) k k! (x – x0)

(3)

with the Taylor polynomials its partial sums, formally, regardless of convergence at any x ≠ x0. It is called the Taylor series of f about (i.e. centered at) x0. Recall that the series (3) but communicates a sequence of terms (3.1.1), that it is the factors of the powers that count for the moment: the Taylor coefficients. ∞

Any positive rational power fr (x) = (1 + x)r owns a Taylor series Σk=0 ak xk where, in accord with Newton’s formula 3.2.3(4), ak =

f

(k)

r(r−1)· ... · (r−k +1) r (0) (1 + 0)r−k = ( k ). k! k! =

Newton would not have struggled as much with the binomial coefficients if he had revealed some analytic feature of his power series a bit earlier than he did.{1, 2a} The Taylor series of such a function fr converges to fr (x) if –1 < x < 1. How about extending (3) to “f (x) = …” in general? We might ask like follows. Let a function f on ℝ have a Taylor series (3) that converges everywhere; would some neighborhood (x0 − δ, x0 + δ ) of x0 then exist where the values of (3) equal those of f (x)? The answer is no, being substantiated – a century later – by Cauchy’s famous counterexample{2b}: a function f, depicted on {2c}, with derivatives on ℝ of any order and a Taylor series about x0 = 0 which, though (trivially) convergent on all of ℝ, does not “function” in the sense that it does not converge to f (x) anywhere except for x = 0 (Appx. C-3). This motivates the following definition: Existence providing, the Taylor series (3) of some function f on D is called a Taylor expansion of f on any set S ⊆ D\{x0} where it converges to f (x). It proves great a distinction for a function when owing a Taylor expansion; to realize how great, we must wait for what was initiated by Cauchy (3.4.1) and accomplished afterwards. * * *

3.2 The power of power series

33 ∞

Newton had regarded the relation f (x) = Σk=0 ak xk as a problem of this kind: Given f on a domain including x = 0, what is the power series, how is f to be expanded? He would not ask whether such expansion exists at all and never would start out with the right hand side and inquire into the properties of a convergent power series. Those issues only became accessible much later. Here is some outlook. A basic proposition reads thus: If Σ ak converges then Σ|ak xk | and hence Σ ak xk converge whenever – 1 < x < 1.

(4)

For the proof, one will profit from the hypothesis only in that, with some B, there is |ak | ≤ B for all k. As

Σnk=0 |ak xk | ≤ B Σnk=0 |x|k ≤

1

B 1 – |x| , n = 0 , 1 , … ,

the series Σ|ak xk | converge by virtue of the least upper bound property of real numbers (see 4.3). In order to make convergence of Σ|ak xk | infer convergence of Σ ak xk the Cauchy criterion is required both ways. Thus, saying that “an absolutely convergent series converges” is not a matter of linguistics (Appx. C-4). Following the proof of (4), we get in general: If a ∞

series Σk=0ak (x – x0)k converges at x1 ≠ x0 then it converges absolutely for all x that are closer to x0 than x1, since | x – x0 | < | x1 – x0 | provides boundedness of the sequence k

Σnk=0 | ak (x – x0)k | = Σnk=0 | ak (x1 – x0)k | | xx1 –– xx00 | . From the above we can draw a complementary conclusion. If a power series Σ ak (x – x0)k diverges at some x2, it must diverge at any x where | x – x0 | > | x2 – x0 |. Whenever there are points x1 ≠ x0 of convergence as well as points x2 of divergence, then some r > 0 exists such that the series converges on the open neighborhood Nr (x0) ≔ {x : | x − x0 | < r } of x0 and diverges at any x with | x − x0 | > r, leaving open what happens at the endpoints. In that case, Nr (x0) is named the series’ interval of convergence and r its radius of convergence (why “radius”, see below). It is common use to make r = ∞ indicate convergence on all ℝ. In those days, power series intuitively were supposed to share all analysis with polynomials (3.2.3). There is much indeed. We are going to list some features which will be granted firm footing no sooner than in late 19th century (4.2, 4.3). The arguments readily apply to series Σ ak(z – z0)k of complex valued terms, with the Euclidean distance | z | = | x + i y | ≔ √( x2 + y2) defining neighborhoods in ℂ, the plane of complex numbers. If we have convergence at some, but not all z ≠ z0 then some circle’s boundary separates open areas (3.4.1) of convergence and divergence. For the Taylor expansion of 1/(1 − z) about z0 ≠ 0, Fig. 3.6 depicts the circle of convergence with | 1 − z0 | the radius of convergence (Appx. C-5). In real analysis, a convergent Taylor series is no guarantee for a Taylor expansion; complex analysis makes a large difference: see 3.4.1.

34

3 Series of functions: the prototypes

z0 0

1 Fig. 3.6. On Taylor expansion of 1− z about z0: the Taylor series’ circle of convergence.

1

We revert to the real number line. Convergence provided, the power series in (5) below generates some function f: x↦

∞ Σk=0 ak (x – x0)k, x∈Nδ (x0) with some δ > 0.{3a}

(5)

The series turns out to share the properties (1), (2) of polynomials: On Nδ (x0), the function has derivatives of any order which are obtained by differentiating the series term by term; hence, (1) and (2) get generalized by ∞

f (k)(x) = Σj=k j (j − 1) … (j − k + 1) aj (x − x0)j−k =

(j+k)! Σ∞ aj+k (x − x0)j , x∈Nδ (x0), j=0 j!

(6)

(as to sum notation see Appx. C-2) and f (k)(x0) = k! ak + 0 + … + 0 , k = 0,1, … .

(7)

Thus, the series in (5) is the Taylor expansion of the function f on Nδ (x0). In short: A convergent power series is the Taylor expansion of the function it generates. ∞

Given a representation f (x) = Σk=0 ak xk, x∈Nδ (x0). Then (7) asserts: not a single coefficient must be changed in this equation. This may be rephrased as follows. Suppose that ∞ ∞ Σk=0 ak (x – x0)k = Σk=0 bk (x – x0)k,

x∈Nδ (x0) with some δ > 0;

then, since both series define the same function f on Nδ (x0), we get ak = f (k)(x0) / k! = bk , k = 0,1, ... . Convergent power series (and hence polynomials) with a common center can only be equal if they are identical. (Identity theorem for power series.{3b}) * * * It’s time for an example of power series in action. A differential equation y'' + y = 0, when considered a problem rather than a statement on a function y(x), asks for some function that fits it on a domain like an interval, called a solution. If, trustfully or not, we assume a solu∞

tion y(x) = Σk=0 ck xk, | x | 0, when formally expanded reproduces f (x) and can hence be considered Euler’s finite expression. Its values φ(x) serving as substitutes for f (x) suggest φ(1) = f (1) for candidate of the value wanted. It required an intricate argument to plead for φ(1) as being the limit of an appropriate series.{2h} In {6b} we read that “the most different methods of summation, the calculus of differences, the Calculus, the expansion by chained fractions all lead to the same value”; actually, on pages {3c}, about the same. (Even Gauß gave a value to the series (4)…) * * * Later, the approved value of (2) was questioned in regards to uniqueness. The first to do so was Nicolaus Bernoulli, but Euler kept sure about it{2i}. And he would have been after more

3.2 The power of power series

45

than forty years when somebody found out that the series (2) also specializes from the right of 1+x 0 2 3 5 1 + x + x2 = x − x + x − x ± … ,

(5)

thus claiming Grandi’s series to be valued by ⅔ “as well”. (Indeed, the argument was to produce an infinity of values.{2i}) Euler would have justly objected{2j}: The series in (5), when considered a proper power series, specializes by 1 + 0 − 1 + 1 + 0 − 1 − + ... which, a “gap series” in modern terms, differs from (2) in principle (cf.{6a}). Useless sophistication? That kind of series will play its part in summability. * * * Was it because Euler could not resist the beauty of symmetry when he wrote ∞ k ∞ ∞ Σ–∞ x = Σ0 x −k + Σ1 x k

x

x

= x−1 + 1−x = 0 {4j}

which at best is nothing but a stark form since not a single x assigns an “arithmetic” meaning to both the series in the middle. Nevertheless, the algebraic view on infinite series is a fully legal one. By intuition, Euler was convinced that every series is capable of more than fitting the rules of algebra. There should be arithmetical substance inherent to being exploited. Euler might well be supposed having assumed something to be disclosed some day. In part it actually was.{1k,l} There is no lesser an advocate of Euler’s than Roger Penrose.{8} Much “new-found-land” of analysis was detected on a risky voyage.{6c} Right after Euler’s century, divergent series would largely be suspected close to devilry, and indeed there had been instances of hell being raised. Among what Euler had accomplished through those means much could be interpreted some way to stand the test of modern rigor{2k}, and there is scarcely anything that could not do it any way. {1}

{2} {3} {4} {5} {6} {7} {8}

JAHNKE [1]: {1a} 131 ff. {1b} 133 bottom. {1c} 150 f. {1d} 151 (4.35). {1e} 152 top. {1f}154 f. {1g}135. {1h}154. {1i} 155 below. {1j} 156. {1k} 156 f. − − {1l} [2] 69 f. HARDY: {2a} 13. {2b} 3 (1.2.21). {2c} 2 f. {2d} 13 f. {2e} 8. {2f} 15, 26,40 note § 2.4. {2g} 26 f. {2h} 27, end of 2.4. {2i} 14. {2j} 14 f. {2k} 349 ff. EULER: {3a} [1] 177 ff; [3] 137 ff. {3b} [4] 589. {3c} [4] 587, 594, 596–617. REIFF: {4a} 118 f. {4b} 65. {4c} 66–69. {4d} 57. {4e} 129. {4f} 123 (Latin), 124. {4g} 69. {4h} 129 bottom. {4i} 123. {4j} 100 f. KNOPP [1] 237 ff; [2] 163 f. SPEISER: {6a} IX. {6b} X. {6c} IX below. HEUSER [2] 683. PENROSE [1] 78 f; [2] 122–124.

46

3 Series of functions: the prototypes

3.3

The power of trigonometric series

3.3.1

Facing antipodes

Jean Baptiste Joseph Fourier named it a trigonometric series, the one that is composed by the twin of functions cosine and sine in a characteristic way (see {1b, 2a, 3a}, e.g.). We will call it a Fourier series only in case the coefficients originate from a given function. Often, one finds this difference neglected; it compares with the Taylor series, yet not quite so since every convergent power series performs as a Taylor series (see 3.2.4(5),(7)). Analysis would not be what it is today without the enormous impact of trigonometric series in various fields, without the challenge of their fundamental problems. Initially, trigonometric series were pushed from the outside, afterwards most interest came from mathematics itself. “Difficult and delicate” as those issues are, they “caused”, in the words of Walter Rudin{1a}, “a thorough revision and reformulation of the whole theory of functions of a real variable”. In many regards, trigonometric series differ strikingly from power series. The latter have a long tradition. We pointed out the special cases which Archimedes (1.3) and Oresmus (2.1) had considered; Nīlakantha made the terms vary. Newton’s Calculus leaned upon them in general (3.2.3), for good reason. They form the most regular functions; by definition, the regular function is the one that allows for representation by an ordinary power series (see 3.2.4, also 3.4.1). Fourier studied erratic processes in nature and, as others before, he noticed “irregular” behavior when analyzing series of trigonometric functions. Strangely enough, these functions are regular themselves, are arranged in a regular manner – and nevertheless prove capable of representing discontinuity! In applied mathematics Fourier’s series started an enormous career – which was to even be outdone in pure mathematics. {1}

{2}

{3}

3.3.2

RUDIN: {1a} [1] 171; [3] 217 f. {1b} [1] 170–181; [3] 216–224. {1c} [1] 170 f; [3] 216 f. {1d} [1] 158 f; [3] 201 ff. {1e} [1] 178 Ex.1; [3] 229. {1f} [1] 172. {1g} [1] 176. KNOPP: {2a} [1] 360 ff; [2] 350 ff. {2b} [1] 361; [2] 350. (Cf.{1c} for a0.) {2c} footnote: [1] 177; [2] 175. {2d} [1] 177; [2] 175 f. {2e} [1] 365 f; [2] 355. {2f} [1] 366; [2] 356. POWELL; SHAH: {3a} 97–149. {3b} 102, Def.5.6. {3c} 121.

Euler evaluating a trigonometric series

Decades before Fourier did so, Euler focused on series like k+1 sin kx Σ∞ k=1 ( −1) k , − π < x < π.

(1)

In case that x = π/2, where sin 2jx = 0 and sin (2j − 1) x = ( −1)j+1 ( j = 1,2, ...), this series becomes Leibniz’ series 3.2.1(1), with its value π/4 matching Euler’s “valuation” x/2 of (1) (cf. 3.3.4{3d}).

3.3 The power of trigonometric series

47

Euler starts out by what the terms of (1) suggest: he makes them play the role of primary functions{1a}: sin k x = k

x

∫ 0 du cos ku

=

x

∫ −π du cos ku ,

− π < x < π.

Only the first integral will work. Regardless of analysis, Euler puts ∞ sin kx Σk=1 (−1)k+1 k

=

∞ ∫ 0 du Σk=1 ( −1)k+1 cos ku x

≡

x

∫ 0 du φ(u),

− π < x < π,

(2)

where the integrand φ(u) denotes a series that diverges on (− π, π){1b}. In two ways of valuation, Euler assigns φ(u) a constant value. In the sequel, we will comment on what the subject of {2a} is and how it refers{2b} to {3a}. (For Euler’s alternative see {1c, 2b, 3b}.) The integral on the right of (2) is made subject to substitution (cf. {2c}) such that φ(u), − π < u < π, takes the form ∞

∞

φ(t − π) = Σk=1 ( −1)k+1 cos k(t − π) = Σk=1 ( −1)k+1 (cos kπ)(cos kt) ∞

= 1 − Σk=0 cos kt, 0 < t ≔ u + π < 2π,

with likewise divergent series{1b} which Euler valuates (cf. {3a}, middle and bottom) through

Σ∞ k=0 cos kt

1

∞

= Re Σk=0 eikt = Re 1 − eit , 0 < t < 2π,

where eit ≠ 1 (wheras 1 ≡ |eit|k ). Now, 1 + eit e−it + 1 1 + cos t − i sin t 1 it = 2it = −it − 2i sin t 1−e 1−e e − eit = with

∞ 1 its real part (equal to Grandi’s valuation of 3.2.6(2) alias Σk=0 (eiπ)k ). So, (2) results in 2 k+1 sin kx Σ∞ k=1 ( −1) k

=

x

∫ 0 du (1 − 12 )

x

= 2 , − π < x < π. {2d, 3c,d}

(3)

When extended to all x ∈ℝ, the series (1) defines the 2π-periodic function s with s(x) = x/2, x∈(− π, π), s(π) = 0, whose graph is Fig. 3.9, being called the sawtooth. s(x) S 2

S

S

x Fig. 3.9. Euler’s saw.

We might suppose (1) to have been one of the inspiring series that caused Euler and others to develop the concept of what later would be called the Fourier series and expansion. At

48

3 Series of functions: the prototypes

present, this is illustrated by how the function s and the coefficients ( −1)k+1/k of the trigonometric series (1) relate: j+1

π

π

(−1) ∫− π dx 2x sin kx = Σ∞ j=1 j ∫− π dx (sin jx)(sin kx) =

=

(−1)k+1 π 2 k ∫ − π dx sin kx

(−1)k+1 k π , k = 1,2, … .

(4)

For the moment we note, by definition (cf. 3.3.4): (4) renders the series (1) the Fourier series of s, (3) is the Fourier expansion of s on (− π, π). * * * How could Euler have been sure about such a striking discovery? His fundus provided support. The terms in (1) when integrated point to Euler’s legendary enterprise with the squared unit fractions (3.2.6(1); Appx. C-8). By his result (3) Euler obtains k+1

x

k

(−1) (−1) Σ∞ ∫0 dυ sin kυ = Σ∞ k =1 k k =1 k2

x2

(cos kx − 1) = 4 , − π < x < π.

(5)

On the right, Euler arrives at a correct equation that remains valid for x = π (likewise as does 3.2.1(2) for x = 1, due to Abel’s limit theorem, Thm.4.1. Since cos kπ = ( −1)k, there comes 2k

(−1) Σ∞ k =1 1 Σ∞ k =1 k2

=

− (−1)k = k2

2 Σ∞ j =1 (2j − 1)2

π2

= 4,

1 1 ∞ Σ∞ j =1 (2j − 1)2 + Σj =1 (2j)2

π2

1

∞ 1

= 8 + 4 Σj =1 j2 , hence

∞

Σk=1 k12

π2

= 6.

Euler’s great result marks a steep peak in analysis. The first access to it was called an evaluation in 3.2.6 according to Appx. C-8. The present access largely makes use of divergent series, thus succeeding by what we called valuation. Both routes were ridge walks, risky before climbers became equipped with the tools of modern analysis. It is not only we who judge the latter more doubtful. Euler would have thought it to be confirmed by the first of his proofs – a further plea in favor of divergent series! For lack of Archimedean proof, the pre-modern analysts largely had to rely on their being convinced. Euler’s conviction grew out of what he experienced when he acquired the same by various arguments. In 3.2.5 and 3.2.6 we had a look into Euler’s workshop and learned about his open mind. How strong a need did he himself feel to have such “algebraic” results been reconfirmed by so-called “arithmetic”? Analysis that is. Today we know he didn’t have much of a choice. Anyhow, he could well trust in his intuition. {1} {2} {3}

BURKHARDT: {1a} 858 top. {1b} 832 (25). {1c} 830–832 (23), (25). {1d} 826 f. HARDY: {2a} 2 f. {2b} 21 § 1.2. {2c} 2 (1.2.5/3). {2d} 3 (1.2.18). EULER: {3a} [4] 582. {3b} [5] 445 f. {3c} [4] 584. {3d} [5] 446.

3.3 The power of trigonometric series

3.3.3

49

Priming problems

Good theory needs good problems. Here is one from outside the world of theory.{1a, 2a, 3, 4a} Consider a string stretched like Pythagoras’ monochord. When plucked by thumb it starts from a kink forming a triangle over its basic position; the very moment it goes into motion the shape becomes extremely smooth. The motion-picture is to be described by analysis. Its theory says that the transversal displacement u(x, t) at the point x ∈[0, π] of location and at the point t ≥ 0 of time has to obey, in principle, the partial differential equation ∂2 u ∂2 u = ∂ t2 ∂ x2

{5a}

(1)

(equating to within a constant factor, in reality, or else being a matter of choosing the units). On {6a, 7a} we learn how to find a general solution of (1) and how it is adapted to the boundary condition u(0, t) = u(π, t) = 0, t ≥ 0.

(2)

A convenient kind of “ansatz”{6a} (cf. the differential equation in 3.2.4) leads to the functions uk (x, t) = (sin kx)[ak cos kt + bk sin kt], 0 ≤ x ≤ π, t ≥ 0, for k = 1,2, … , {6b}

(3)

which all comply with (1), (2). That extends to their sums and, appropriate convergence provided, to the series they form: U(x, t) =

Σ∞ u (x, t) , k=1 k

0 ≤ x ≤ π, t ≥ 0.{6c}

(4)

See {4b} for interpretation: Standing waves are superposing here, harmonics which by their modes correspond to the scale of pure tones that compose any sound. So far all this concerns the swinging party. It needs a good start. In case this is done by releasing the fixed string without acceleration, the move follows the single initial condition ∞

U(x, 0) = Σk=1 ak sin kx = f (x), 0 ≤ x ≤ π,{6d}

(5)

imposed by a function f that allows for a representation (5). That looks like quite some requirement! How about the chance to make both modes, the start and the move, join “analytically”? How about a function f that has the triangle of a thumb start for the graph? The problem aroused much controversy (for details see {1b, 3}). Daniel Bernoulli{2b} came up with a formal solution which, compared to (4)/(3), reads

Σ∞ a (sin kx)(cos kαt), k=1 k

0 ≤ x ≤ π, t ≥ 0.

It had been opposed by Euler who explicitly refers to (5) when saying “... consider ... a string which, before release, has a shape which cannot be expressed by the equation [(5) that is]”{4c}. To him, f in (5) need to be a single analytic expression, a function à la Euler rather than a junction of distinct ones with graphs in the place of one graph.{4d} He called Bernoulli’s argument a “reductio ad absurdum”{2c} and would ever stick to the credo he shared with d’Alembert: Every function has a graph, but not every graph represents a function.

50

3 Series of functions: the prototypes

Decades later, Fourier considered a process in nature that can be initiated most arbitrarily: the propagation of heat. When piecing together bodies on different temperatures the initial function will violate even continuity{4e} – whereas a string would at most start by kinks. Anyway, Bernoulli and Fourier were on the right track. Long after Fourier had presented a memoir to the French Academy on the “Theory of Heat”{2d}, in 1807, his La Théorie Analytique de la Chaleur came out in 1822{2e, 7b, 8}. The ideas are being called a “master stroke” in {4f}, though La Théorie shows that theory was not exactly the author’s domain. Others{2f}, including Cauchy{1d}, would try their hands at mathematical treatment, without satisfying results. Ultimately, Peter Gustav Lejeune Dirichlet does accomplish the breakthrough when, in 1829, he sets forth a subtle theory“Sur la convergence des séries trigonometriques” and furnished rigorous proofs.{1e, 2g} {1} {2} {3} {4} {5} {6} {7} {8}

RIEMANN: {1a} 228–232. {1b} 229–232. {1c} 235. {1d} 234. {1e} 235. HOBSON [2]: {2a} 476–479. {2b} 477 f. {2c} 478. {2d} 480. {2e} 480 f. {2f} 481. {2g} 482. GRATTAN-GUINNESS 2–13. DAVIS; HERSH: {4a} [1] 257 ff; [2] 268 ff. {4b} [1] 258 f; [2] 269. {4c} [1] 259; [2] 269 f. {4d} [1] 59; [2] 270. {4e} [1] 261; [2] 272 f. {4f} [1] 263; [2] 274. HOUSTOUN: {5a} 115–117. {5b} 115 f. {5c} 116. HEUSER [2]: {6a} 118–120. {6b} 120 (132.5). {6c} 121 (132.8). {6d} 121 (132.9). SONAR: {7a} 475 f. {7b} 477, 480. RUDIN [1] 171; [2] 217.

3.3.4

The mysterious . . . Fourier series

We proceed as Euler and others did and as did Fourier, thirty years later, unaware of Euler’s initiative (cf. 3.3.2(4)).{1a} The latter was the first to answer the fundamental question: Given, with real quantities, ∞

f (x) = a cos 0x + Σk=1 [aj cos jx + bj sin jx], – π ≤ x < π;

(1)

how does the function explicitly relate to the coefficients a, aj , bj? (We will keep to the period interval [– π , π) and could as well take any [ξ, ξ + 2π) or (ξ, ξ + 2π], ξ∈ℝ.) The answer was, eventually, found by a trick: When being multiplied by cos kx , sin kx (k = 1,2, ...) and integrated formally, equation (1) readily yields the Euler-Fourier formulas{2, 3a, 4a, 5a} π

π

∫ –π f (x) dx = a ∫ –π cos 0x dx = a · 2π ≕ a0 π ,

(20)

π ∫ –π

(2a)

f (x) cos kx dx =

π

π ak ∫ –π

cos2 kx dx = ak π (k = 1,2, …),

π

∫ –π f (x) sin kx dx = bk ∫ –π sin2 kx dx = bk π (k = 1,2, …)

(2b)

on the grounds that, for any j, k = 1,2, … , there holds π

π

π

∫ –π1·cos kx dx = ∫ –π 1·sin k x dx = ∫ –π (cos jx)(sin kx) dx = 0, π

π

∫ –π(cos jx)(cos kx) dx = ∫ –π(sin jx)(sin kx) dx = δjk · π,

(3a) (3b)

3.3 The power of trigonometric series

51

where δjk = 0 if j ≠ k and δkk = 1. Only after quite some time, Fourier as well as Euler hit upon (3a,b) whereby, at a stretch, such a series of integrals shrinks down to but one term.{1a} n By analogy of (3..) to the scalar product in a vector space ℝ , the functions cos 0x, cos 1x, sin 1x, cos 2x, sin 2x, … are said to be orthogonal to each other and to form, moreover, an orthonormal system after being multiplied, in this order, by 1/√ (2π), 1/√ π, … .{4a} (Appx. C-9; as to norm, cf. 4.4.) “Orthogonal” is also used to denote the kind of series in (1). In 4.2.3 a condition on (1) is pointed out sufficient to make (2..) result properly. By the time when this was assured the basic features of the Fourier theory had long been established. Provided the integrals above do exist in some appropriate sense – without risk we may assume Riemannian integrals (“R”-integrals) throughout – those numbers a, ak, bk are called the Fourier coefficients of f. The series they form is called the Fourier series assigned to f. Regardless of convergence.{3b} To point out the latter, one writes “ f ~ ...” rather than “ f (x) = ...”. (For other period intervals see Appx. C-10.) A trigonometric series (1) is a Fourier series only if Σ ak2 < ∞, Σ bk2 < ∞.{3c} Trigonometric series are easier to handle when given in terms of the system e0, eix, e−ix, e , e−2ix, … which likewise is called orthogonal in that 2ix

π

π

∫ –π eimx e−inx dx = ∫ –π ei(m−n)x dx = δmn· 2π (m, n ∈ℕ0) and orthonormal after being normed by 1/√ (2π) throughout.{4b, 6a} (Note eiπ = e−iπ.) For translation, observe 3.2.5(8) whereby a0 a = 2 = c0 , ak cos kx + bk sin kx = ak

ekix + e−kix ak − i bk kix ak + i bk −kix ekix − e−kix + b = e + e k 2 2i 2 2

≡ ck ekix + c−k e−kix (k∈ℕ) , ∞

1

π

f (x) = Σk=−∞ ck ekix, ck = 2π ∫ –π e−kix f (x) dx,{6a} ∞

N

where the symbol ”Σn = −∞”, according to how it develops, has to be read “lim N →∞Σn = −N “{6b}. But also see {7} where this series is being regarded a Laurent one (cf. 3.4.2(5)) with the annulus degenerated. Recall that a Taylor series is a power series which is generated by an appropriate function; a (non-trivially) convergent power series itself furnishes such function (3.2.4(5)); thereby, any convergent power series may be called a Taylor series. Things are different with trigonometric and Fourier series (see below). So we have much reason to call a trigonometric series a Fourier series only in the case that its coefficients are related to a function by the Euler-Fourier formulas (20, 2a,b). We speak of Fourier expansion no sooner than when, and nowhere else than where, a function is being referred to such that its Fourier series converges to the value of the function. * * *

52

3 Series of functions: the prototypes

On pursuit of Fourier series, surprise might wait around the corner. A new access to Fourier coefficients is opened by so-called functional analysis, a theory where analysis is applied to “functions as such” in the following sense. The function f: x ↦y itself, rather than the x and the y, is considered a “point”, that is in a “space” together with functions that share a distinct n property. Just like the points u = (u1, ... , un), v = (υ1, ... , υn) ∈ℝ are at the Euclidean distance ½

n

|| u – v || ≔ [ Σν=1 (uν – υν )2] , the integrable functions f, g : [a, b] → ℝ are points in a vector space and get assigned the distance b

½

|| f – g || ≔ [∫a ( f (x) – g(x)) dx] 2

(4)

by the so-called square or quadratic mean{8} of f – g. (See 4.4 for more on || … ||, which clearly generalizes the absolute value | … | on the “space” ℝ = ℝ1.) For f : [– π, π] → ℝ an integrable function we consider the nth partial Fourier sum a

s(n) : x ↦ 20 +

Σnν=1 [aν cos νx + bν sin νx]

according to (2..), a Fourier polynominal with n ∈ℕ0 being fixed for the moment (note an empty sum to mean zero). How do the (real) trigonometric polynomials α

t(n) ≡ t(n)[α0 , α1 , β1 , … , αn , βn] : x ↦ 20 +

Σnν=1 [ αν cos νx + βν sin νx ]

compete with s(n) for approximating f by means of (4; a = – π, b = π)? It turns out that the infimum of the numbers || t(n) – f || is assumed, exclusively, by t(n)[a0 , a1 , b1 , … , an , bn] = s(n). That is to say that 2n+1

|| s(n) – f || = min { || t(n) – f ||: (α0, α1, β1, … , αn, βn)∈ℝ

}, n = 0,1, ... ,{5b, 6c}

(5)

th

characterizes (defines) the n Fourier polynomial of f. Among all trigonometric polynomials of degree n it is this one that, by Euclidean distance, proves closest to f ! (Mathematics is beauty-ful.) * * * The function s(x) depicted in Fig. 3.9 is continuous except for the points x* = ± (2k +1) π, k = 0,1, ... . There, the one-sided limits s(x* ± 0) of s(x*+ t), s(x*− t) as 0 < t → 0 exist and are different, indicating a jump discontinuity each. The value s(x*) = 0 happens to lie half way amidst those limits s(x* + 0) = − π/2, s(x* − 0) = π/2, yet not by chance: In like cases, convergent Fourier series will thus behave. We may call a function piecewise continuous if it is continuous except for at most finitely many jump discontinuities on any finite interval of its domain. By virtue of {5c}, which is due to Fejér’s Thm. 5.9 at 5.3.3, we will have the following Theorem 3.1. Let the (2π-periodic) function f be piecewise continuous. If the Fourier series of f converges at x then it converges to f (x − 0) + f (x + 0) 2

.

(6)

3.3 The power of trigonometric series

53

For memory’s help: The Fourier series of a piecewise continuous function would rather diverge at x than converge to any value other than indicated by (6). (For thorough expansion, a function f with a jump at x need be granted a fair rest in between which is provided if f (x) is defined by (6).) When f is continuous at x then (6) equals f (x) and we may note the corollary Theorem 3.2. If the Fourier series of a (2π-periodic) continuous function converges everywhere then it is the Fourier expansion of the function. However: See below! * * * In order that a 2π-periodic function be assigned a Fourier series, it need only give the integrals (2..) a meaning. This was out of the question with piecewise continuous functions. When Bernhard Riemann engaged in Fourier series the first he did was create a clear and more general concept of the definite integral.{9} (The R-integral admits countably many discontinuities on the bounded interval of integration.{4c, 10}) Soon after, Riemann’s integral became generalized by Henri Lebesgue{4d, 5d, 6d} to better comply with the needs of advanced analysis, in particular of Fourier theory.{4e,f, 5a, 6e,f} Any kind of integral is not being affected if the values of the integrand get changed to a definite extent depending on the type of integral. The Fourier series of an R-integrable function f would not take notice if, say, finitely many values f (x) were manipulated.{6g} * * * We point out some further peculiarities of trigonometric series. Where uniqueness is concerned, k+1 sin k x Σ∞ k=1 ( −1) k

π

∞ sin k x

= 2 − Σk=1 k

, 0 < x < π,

(7)

exemplifies differing trigonometric expansions of the function x/2. Both sides of (7) display Fourier expansions on (0, π) (in the sense we defined expansion at 3.2.4): On the left hand in view of 3.3.2(3), on the right due to the Fourier expansion of (π − x)/2 on (0, 2π){3d} which, by the way, allegedly was the first one in the world, discovered in 1744. It is an open question whether, on a common period interval, a function may have different trigonometric expansions.{3e} (Appx. C-11.) Judging mainly by various experience, Fourier held that “arbitrary functions” allow for expansion (which at best meant piecewise continuous ones). On this, he was given full credit by Cajori{11a} as well as by the notable historian Dirk Struik who maintains that Fourier “fully cleared the situation”{11b} − despite an early counterexample, a continuous function that is!{3f} Philip Davis and Reuben Hersh ascertain that “Fourier … neither stated nor proved a correct theorem about Fourier series.”{1b} (8) Even if Fourier’s credo may have been questioned in his day, continuity might well have been considered a warranty for Fourier expansion. Who would imagine a function f continuous at x and its Fourier series not converging to f (x)? As late as 1873, such a function was found.{3g} In the sequel, optimism was lowered much further. There is, for instance, no an-

54

3 Series of functions: the prototypes

swer so far to the question: Is it true that, for every continuous function, there exists some point x at which the Fourier series of the function converges?{6g} For what reason should Fourier’s series be loyal to its generator, the integrable function? Specific reasons were laboriously detected. After furnishing a necessary and sufficient condition for convergence to the value (6){3h}, Dirichlet and others developed sufficient “rules”{3i}. Indeed, to own a Fourier expansion with whatever suitable type of integral (2..), this seems to be a property of a function that cannot equivalently be reduced to any from the catalogue of analysis. If that be true, the existence of a Fourier expansion has to be considered a fundamental property of a function.{3j, 5e} * * * People keep looking for well rounded results, considered “natural”. The first one was achieved by changing the point of view. All those previous endeavors had to do with how the Fourier series relates to the function’s values at single points. The distance (4) of functions makes global approximation possible in that (5) assures || s(n) – f || → 0 as n → ∞ {5f}

(9)

by the name of convergence in square (or quadratic) mean.{8} * * * The portrait of Fourier was that of a bold experimental mathematician. As to mathematics, see (8) above, a quotation that is thus to be completed: “Fourier was right even though he neither ...”. Right in matter when he refuted the credo of Euler and d’Alembert, right in the sense of a vision. Besides what was mentioned about trigonometric series in pure analysis (3.3.1), they are just as important for science. By virtue of orthogonal systems in general, orthogonal series were conceived to serve specific problems of expansion. When functions and coefficients of trigonometric series were related as was sketched in 3.3.3, waves became considered the composition of elementary waves, thus being “analyzed”. It marked the starting point of a vast modern discipline by the name of Fourier analysis. Among the many merits of Fourier expansion, there is another fundamental one: today’s notion of function. Euler made divergent series define a function, yet would not accept the saw shown in Fig. 3.9 to represent one function. To him a single analytic expression was required to make a function. Well, the Fourier series is a single one. It helped open the minds to accept Dirichlet’s liberal concept, allowing for the definition “ f (x) ≔ 1 if x is rational and f (x) ≔ 0 else ”, a function whose graph denies depiction.

{1} {2} {3}

{4}

DAVIS; HERSH: {1a} [1] 262; [2] 273. {1b} [1] 263; [2] 274. HOBSON 480. KNOPP: {3a} [1] 363 f; [2] 352 f. {3b} [1] 365; [2] 355. {3c} [1] 372; [2] 362. {3d} [1] 387; [2] 375. {3e} [1] 366, Nr.4; [2] 355, no.4. {3f} [1] 391 bottom; [2] 379 “1.”. {3g} [1] 391, footn.1; [2] no.216. {3h} [1] 371 f; [2] no.203. {3i} [1] 376–384; [2] nos.206–208. {3j} [1] 365 f; [2] 355. APOSTOL: {4a} 309, 307. {4b} 307. {4c} 171. {4d} 259–261. {4e} 309 f. {4f} 311.

3.4 Cauchy

55

HEUSER [2]: {5a} 124 f. {5b} 129 f. {5c} 158. {5d} 84 ff. {5e} 154. {5f} 163–165. RUDIN: {6a} [1] 171 f; [2] 17. {6b} [1] 170; [2] 217. {6c} [1] 172 f; [2] 219. {6d} [1] 227 ff; [2] 253 ff. {6e} [1] 252 ff; [2] 223 f. {6f} [1] 255; [2] 388. {6g} [1] 176. {7} PENROSE 170 f. {8} KOREVAAR [1] 239 bottom, p = 2. {9} RIEMANN [1] 239. {10} KÖRLE 205–208. {11} {11a} CAJORI [1] 270. {11b} STRUIK [1] 168 top; [2] 160. {5} {6}

3.4

Cauchy

3.4.1

Series of powers in the hands of Cauchy and Laurent

Augustin Louis Cauchy claimed being the one to re-establish rigor in analysis.The notion of limit is the point from which to start analysis. Cauchy was sure to have made it a fixed point. General Nicolas Bourbaki – a pseudonym for a more recent and ever renewing group of leading French mathematicians – makes the point: “La notion de limite, fixée une fois pour toutes, ... ”{1}. Once and for all. When lecturing, Cauchy felt responsible for his theorems to be grounded on firm notions. The students would not reward it, yet his“Cours d’Analyse”{2a, 3a} was to become a long lasting incentive. By diverse kinds of impact, Cauchy greatly advanced analysis, regardless of deficiencies due to the lacks in the foundations of analysis. In the latter respect, he does not compare with a contemporary, the Bohemian clergyman Bernhard [Bernardus] Bolzano{3b, 4} (cf. 4.1). The pretentious standards Cauchy thought to have accomplished in analysis were to be attained only later in the course of the 19th century. Aside from this point, and also from an abundance of successful enterprises in various fields of mathematics and of science, Cauchy made a most fundamental contribution to analysis: his theory of functions was to become the Theory of Functions. Euler{3c} and Cauchy opened the scene of action that calculus had been waiting for; the actors were the complex numbers z. Extending ℝ, these form a structure denoted ℂ with a “distance” | z | from z = 0 that generalizes the absolute value of the reals. The construct proved to be the appropriate frame for differentiation and non-Newtonian integration, an Eldorado despite Cauchy’s judging on √(−1) that“one does not know what this supposed sign signifies nor what sense to attribute to it”{5}. Like the negative numbers, the complex ones had to struggle for acceptance. Confidence in the number “i” would not grow until it was granted geometrical asylum in the number plane ℝ2 where z ∈ ℂ is uniquely represented by z = x + iy through coordinates x, y ∈ℝ (Fig. 3.8). That “complex number plane”, conceived around 1800{6} not by Gauß alone, came to be noticed by the public only decades later. * * * It might be worthwhile to detail some basic notions of complex analysis that are referred to (cf. {7, 8, 9, 10, 11, 12}, e.g). We consider single-valued functions of one complex variable. Their domains or subdomains D ⊆ ℂ will be considered open in that around every z* ∈ D there exists

56

3 Series of functions: the prototypes

some “open” disk d(z*; r) ≔ {z ∈ℂ : | z − z*| < r} ⊆ D, 0 < r ≤ ∞. All those notions from real analysis as limit, continuity and even the differential quotient can readily be translated into complex analysis. A function f is called differentiable at z* if there exists c ≕ f '(z*) with the property

| f (z z) −− zf*(z*) − c | → 0 n

n

whenever | z n − z* | → 0 , z* ≠ zn ∈D.

(This means a lot more than f (x + iy) to be partially differentiable with respect to x and y !) See further denotations: A function f is analytic on D if it is differentiable at any z* ∈D {7a, 8a}. It is analytic at z* if being analytic on some disk d(z*; δ) ⊆ D; otherwise, f is said to have a singularity at z* which is an isolated one if, moreover, f is analytic on some pierced (or deleted) disk d·(z*; δ) ≔ {z ∈ℂ : 0 < | z − z* | < δ} ⊆ D. In the “real” world of calculus, integration on an interval may be interpreted “ from a to b ” or vice versa. As of now, continuous functions will accordingly be integrated along a path{7b, 8b, 9a, 10a, 11a} (or oriented curve) within its domain D. Strictly speaking, a path is a class of “appropriate” mappings each of which represents it like, for instance, the two mappings t ↦ (cos t, sin t), 0 ≤ t ≤ 2π, and t ↦ (cos 2t, sin 2t), 0 ≤ t ≤ π, differently describe a point moving counterclockwise along the contour of the unit circle for one time. Let t ↦ γ(t) ∈D, a ≤ t ≤ b, denote a path in D; in case of γ(a) = γ(b) the path is closed. Given subdivisions a = t0 ≤ t1 ≤ ... ≤ tn = b of [a, b], the curvilinear integral of f along γ is to be approximated by

Σnν=1 f (γ(τν)) (γ(tν) − γ(tν−1)) = Σnν=1 f (γ(τν)) γ(tνt)ν −− tγ(tν−1ν−1) (tν − tν−1),

tν−1 ≤ τν ≤ tν,

and becomes defined through

∫γ f (z) dz ≔ ∫ ba

f (γ(t)) γ'(t) dt =

∫ ba Re[ f (γ(t)) γ'(t)]dt

b

+ i ∫ a Im[ f (γ(t)) γ'(t)]dt,

(1)

an essential being that the value of the curvilinear integral does not depend on how the path is represented: on how it is “parametrized”. * * * With regard to differentiability, real and complex analysis differ greatly. How does a real function φ, differentiable on some interval (− δ, δ ) ⊆ ℝ, compare in general with a complex

function f that is differentiable on some disk d(0; δ) ⊆ ℂ ? For one thing, there need not exist φ"(0) (as shows φ(x) = − x2 if x ≤ 0, φ(x) = x2 if x > 0), let alone a Taylor series about x0 = 0. (Cauchy’s example (cf. Appx. C-3) shows a Taylor series to exist without yielding an expansion anywhere.) In contrast to φ, the complex function f admits Taylor expansion about every z0 ∈d(0; δ). So, the complex plane is the very soil where the Taylor series can take root. The lively point of calculus on ℝ had been the interaction of differentiation and integration. As concerns ℂ, analogy is truly sophisticated. We only deal with a simple fundamental feature. Cauchy’s integral (or main) theorem in the Theory of Functions asserts: The integral (1) has value zero in case that γ is a closed path inside of which f is analytic.{7c, 8c, 9b, 11b} As in {11b}, we

3.4 Cauchy

57

will consider simple closed paths on contours like circles, i.e. where γ(t1) ≠ γ(t2) if a < t1 < t2 < b (yet cf. {9c}). What may happen to (1) when there are singularities of f inside the path? * * * To illustrate Cauchy’s main theorem we choose the function 1 g(z) = 1− z , z ≠ 1. It is called the analytic continuation of the geometric series in the following sense: g is the ∞

only analytic function on ℂ \{1} such that g(z) = Σk=0 zk holds on whatever a disc inside of d(0; 1). (See {8d, 10b, 11c}, e.g.) Let z0 ≠ 1, ρ > 0. The mapping γ[z0|ρ]: t ↦ γρ(t) = z0 + ρeit (0 ≤ t ≤ 2π) represents a circular path that leads counterclockwise from the point z0 + ρ back to it. If and only if ρ < | 1 − z0 | then only points are encircled at which g is analytic. (Fig. 3.6. shows a maximal disk. It may be imagined to develop when a 2-dimensional balloon with contour γ[z0|ρ], ρ < | 1 − z0 |, is blown up until it hits upon the singularity of g at z = 1.) The series in 1 1 z − z0 −1 g(z) = (1 − z ) − (z − z ) = 1 − z ( 1 − 1 − z ) = 0 0 0 0

1 Σ∞ (z − z0)k k=0 (1 − z0)k+1

has the radius of convergence r≔ | 1 − z0 | and converges on any circle line {z : | z − z0 | = ρ}, 0 < ρ < r, such that term-by-term integration is granted in

∫γ[z0|ρ] g(z) dz

1 Σ∞ ∫2π (γρ(t) − z0)k γρ'(t) dt k=0 (1 − z0)k+1 0

=

∫2π 0 g(γρ(t)) γρ'(t) dt

=

iρ Σ∞ ∫2π ei(k+1)t dt k=0 (1 − z0)k+1 0

=

(2)

k+1

= 0 , 0 < ρ < | 1 − z0 |.

(Cauchy had no problem with this kind of integration, no more than Newton had. See Appx. C-12.) Now, when integrating g along the circular paths γ[1|σ], i.e. around its only singularity, we readily get

∫γ[1|σ] g(z) dz

2π

= −i ∫0 dt = −2πi,

(3)

other than in (2) where there had been no singularities inside of γ[z0 |ρ], ρ < | 1 − z0 |. However, for a curvilinear integral to vanish, the integrand need not be analytic within the path as the integrals of the function 1 h(z) = (1 − z)2 , z ≠ 1, show along any γ[1|σ]: they all result in −it ∫γ[1|σ] h(z) dz = σi ∫2π 0 e dt = 0,

σ > 0.

(4)

58

3 Series of functions: the prototypes

At (2), (3) each, we observed the values of the integrals to be the same for all circles considered. Cauchy’s main theorem in its general version allows for a lot of tolerance where the shape of the contours are concerned. (See the application in the end.) * * * The above functions g and h respectively display poles at z0 = 1 of orders 1 and 2. They specialize from the formal aggregate{7d, 9d}

Σk∞= −∞ ak (z − z0)k

≡ P(z) +

Σ∞ a (z − z0)k, k=0 k

∞

P(z)≔ Σk = 1 a−k (z − z0)−k,

(5)

that splits into the principal part P(z) (= g(z), = h(z)) and a regular part which vanishes in the present case. One may name type (5) an extended power series if not plainly say “power series” (cf. 3.1.2{4}). Those g, h are functions that are analytic on some pierced disk d·(z0; r), 0 < r ≤ ∞, and (trivially) allow for a representation (5) there. Does this happen to any function which is analytic on a pierced disk? Yes, said Pierre Alphonse Laurent in 1843{11d} when he discovered what was known to Weierstraß{11c} before and is hardly due to someone by the name Laurent who, being born in 1841, is given credit in {12}. Let a function f be analytic on some d·≡ d·(z0; r) (which does not exclude it to be analytic on all the disk). Laurent’s theorem {7e, 9e, 10c,d, 11e} ascertains that, in (6) below, the integrals exist, that their values but depend on k and that f (z), z∈d·, is represented by (5) if {10c} and only if {10d} 1 f (z) ak = 2πi ∫γ[z0| ρ] (z − z )k+1 dz, 0 < ρ < r, k∈ℤ, 0

(6)

being called the Laurent coefficients of f about z0. (They are the Taylor coefficients in case that the principal part of (5) is missing.) (Appx. C-13) To its full extent, Laurent’s theorem reads as follows. Let f be analytic on some annulus A ≡ A(z0; r1, r2) ≔ {z : r1 < | z − z0 | < r2} with 0 ≤ r1 < r2 ≤ ∞. Then, (6) with r1 < ρ < r2 generates the Laurent series (5) of f with respect to A{13a} and yields the Laurent expansions on A whose uniqueness is granted by (6) (which sometimes lacks to be pointed out). According to how the wording applies to Taylor and Fourier series, (5) might be named a Laurent series only if it originates from a function through (6); yet, any series (5) is usually called a “Laurent series” if it is not an ordinary power series. In a case like f (z) = (z2 – 1)/(z – 1), the Laurent expansion about z0 = 1 does not display a veritable singularity; the virtual one of the function is to be“removed” by way of defining f (1) ≔ f (1 ± 0). If a function has a proper isolated singularity at z0, then its Laurent expansion about z0, since being unique, serves to characterize the type of singularity{8e, 9f, 10e}. If a−p ≠ 0 and a−p−1 = a−p−2 = ... = 0 then f has a pole at z = z0 of order p; if the principal part P(z) is an infinite series, an isolated singularity of f at z0 is said to be an essential one, as is the case{7f, 9g} with e1/z, z ≠ 0 = z0.

3.4 Cauchy

59

Let us consider Laurent’s theorem when applied to a function analytic throughout d(z0; r). Then, the coefficients (6) follow 3.2.4(7) which makes us arrive at 1 (k) 1 f (z) k! f (z0) = 2πi ∫γ[z0| ρ] (z − z0)k+1 dz, 0 < ρ < r, k = 0,1, … .

(7)

Case k = 0 is Cauchy’s integral formula {7g, 8f, 9h, 10f, 11f} from which (7) is produced by differentiation. Laurent’s theorem gives access to this formula “by way of series”. (Within the complex theory of functions, Cauchy’s domain had been curvilinear integrals. It was Weierstraß who made power series walk in the plane, not before having booted them accordingly.) There is this message in (7): Taylor’s differential quotients only depend on the values f (z) on whatever small a neighborhood of z0; Laurent’s curvilinear integrals only depend on the values f (z) on lines, possibly “far away” from z0. All this indicates how much they cohere, the values of an analytic function! And it marks a triumph of infinite series as well. * * * Uniqueness of the Laurent coefficients makes a further point. Let f be analytic on some d·(z0; r) and (5) its Laurent expansion there. In a certain way, the leading factor of the principal part will turn out a leading actor: it indicates what, if any, is “left over” when closed curvilinear integration is performed about z0. Owing to its role as being “residual” (in Latin, residuum is what remains), Cauchy named the value a–1“le résidu de f en z0”, the “residue of f at z0”. (See, e.g., {2c, 7f, 8g, 9g, 10g, 13b}.) The residues of g, h above had been –1 and 0 by (3), (4), specializing from i(−k+1)t ∫γ[z0|ρ] f (z) dz = Σk∞= 1 a−k ρ−k+1 i ∫ 2π dt 0 e

+

2π Σ∞ a ρk+1 i ∫ 0 ei(k+1)t dt k=0 k

= 2πi a−1, 0 < ρ < r .

(8)

Whenever a closed path of integration encloses only the finitely many singularities z0, ... , zn, the above consideration applies to adequate disks d·(zν; rν). They likewise render residues the sum of which results, save the factor 2πi, in the value of the said contour integral.{2d} There must have been a good reason for Cauchy to coin the notion. Cauchy conceived quite a Residue Calculus{7h, 8h, 9g, 10f, 13c} with surprising applications, namely for evaluating improper integrals in real analysis. To give an idea, here is how it works{8i} in case of ∞

∫–∞ x2dx+ 1 = lim R →∞[arctan R

π

π

− arctan (− R)] = 2 − (− 2 ) = π.

(9)

The function φ (z) = 1/(z2 + 1) has a pole at z = i of order 1. Consider the semicircular paths with radius R which are run by real x from −R to R and by complex R eit, 0 ≤ t ≤ π, from R back to −R , joining to form the closed path Γ [R], say. Choose R ≥ 2. Then, due to variable

60

3 Series of functions: the prototypes

contours in Cauchy’s theorem, any integral of φ along Γ [R] renders the same value as the integral (8) in case of f = φ, z0 = i, ρ = 1. The residue of f at z0 = i is provided by 1

(z − i) φ(z) = z + i = a−1 + a0(z − i) + a1(z − i)2 + … as z → i, hence reads a−1 = 1/(2i). Thus we obtain R π R i eit dx π = 2πi a−1 = ∫Γ [R] φ(z) dz = ∫ –R x2 + 1 + ∫ 0 R2 e2it + 1 dt,

so that R → ∞ makes us arrive at (9) since | e2it + 1/R2 | ≥ | |e2it | − 1/R2 | = 1 − 1/R2 ≥ ¾. * * * It is typical of Cauchy that he made his curvilinear integral also a tool in planar hydrodynamics, its value then meaning energy. Where he could not comply with his goal of settling fundamental analysis, a word of Reuben Hersh{14} is suitable: “Cauchy knew Cauchy’s integral theorem, even though … he didn’t know the meaning of any term in the theorem … … Cauchy had great intuition.” {1} {2} {3} {4} {5} {6} {7} {8} {9} {10} {11} {12} {13} {14}

BOURBAKI [1] 174 middle. CAUCHY: {2a} [3]. {2b} [3] 120. {2c} [1] 306 f. {2d} [1] 307 f. SONAR: {3a} 497, 507. {3b} 497–502. {3c} 454. GRATTAN-GUINNESS 51–57. LÜTZEN 221 f. KLINE [1] 816. BOYER 548. BOTTAZZINI 272. PENROSE [1] 81; [2] 127. APOSTOL: {7a} 434, 458 (Def. 16.32 b to be completed). {7b} 435 ff. {7c} 439. {7d} 455 bottom. {7e} 456 f. {7f} 459. {7g} 443 f. {7h} 460 ff. PÓLYA; LATTA: {8a} 75 ff. {8b} 153 f. {8c} 146. {8d} 231. {8e} 203. {8f} 177 f. {8g} 204. {8h} 207 ff. {8i} 208. LANG [1]: {9a} 94 ff. {9b} 138 ff. {9c} 146 f. {9d} 161. {9e} 162. {9f} 165 ff. {9g} 168. {9h} 145. ASH: {10a} 6, 19 f. {10b} 122 ff. {10c} 74 ff, 3.1.2. {10d} 76, 3.1.3. {10e} 79. {10f} 30. {10g} 83. {10h} 83 ff. HILLE: {11a} 160 f. {11b} 163 ff. {11c} 131. {11d} 209 footn.1. {11e} 209–211. {11f} 175. WUẞING [1] 252. NEEDHAM [2]: {13a} 514 (9.4.2). {13b} 504. {13c} 506 f. HERSH 64.

3.4 Cauchy

61

3.4.2

The Cauchy product

Providing there is convergence of the two series on the left, elementary calculus yields ∞

∞

(Σi=0 ui)(Σk=0 υk) =

∞ Σ∞ (Σk=0 υk) ui i=0

=

∞ Σ∞ (Σk=0 ui υk ) i=0

and the like with the order of summation inverted. However, on the right hand one wants a series which originates from lining up some way the entries of the matrix (ui υk : i, k = 0,1, ... ). How and under what assumptions might this be performed? Cauchy approached the issue via the product of power series. r

s

When multiplied, polynomials Σi=0 ai xi, Σk=0 bk xk of degrees r, s result in a polynomial l l Σr+s l=0 cl x of degree r+s where cl collects all terms ai bk x with i + k = l (0 ≤ i ≤ r, 0 ≤ k ≤ s). The

idea leads up to the formal product of power series ∞ ∞ ∞ [Σ∞ a xi ]·[Σk=0 bk xk] ≔ Σl=0 ( Σ aibk ) x i+k = Σl=0 cl xl, i=0 i i, k ≥ 0, i + k = l l

(1)

l

cl = Σi=0 ai bl – i = a0 bl + a1 bl–1 + … + al b0 = Σk=0 al – k bk = al b0 + al–1 b1 + … + a0 bl , (2) where the entries in the matrix (aibk : i, k = 0, 1, ...) are passed along “slanting lines”, down and up respectively. At (2), the (l+1)-tupels a0, a1, … , al and b0, b1, … , bl meet one another as they would do on a strip of paper that is folded in the middle (Fig. 3.10); so the sums in (2) go by the names folding (the German Faltung), convolution, besides resultant.

a0 a1 … al 1 al |

b0 b1 … bl 1 bl Fig. 3.10. A folding.

* * * Equation (1) is a formal one. It marks the pattern according to which Cauchy defines the ∞

∞

product of any series Σi=0 ui, Σk=0 υk, today called their Cauchy product. It originates from (1) by x = 1 or by renaming ai xi ≕ ui, bk xk ≕ υk and thus reads l l ∞ ∞ ∞ [Σ∞ u ]·[Σk=0 υk] ≔ Σl=0 Σi=0 ui υl – i = Σl=0 Σk=0 ul – k υk .{1a, 2a, 3a, 4, 5a} i=0 i

(The product is commutative, as shows the right hand equation.) Cauchy himself gave an example of two convergent series whose product diverges: see (8) below. That raises the problem: Find sufficient conditions in order that two series convergent to U, V make a Cauchy product convergent to UV. Cauchy first considers “positive” terms and goes straight to prove Thm. 3.3 below. To us, it is worthwhile to primarily look for proof of the following Proposition. Let ui ≥ 0, υk ≥ 0. If

Σ ui < ∞, Σ υk < ∞ then Σ cl < ∞ where

l

cl = Σi=0 ui υl – i.

62

3 Series of functions: the prototypes

Proof. Whenever the matrix terms ui υk take positions within the square of entries (i, k) with i, k = 0, ... , n, then Cn ≔

Σnl=0 Σli=0ui υl – i

n

∞

n

∞

≤ ( Σi=0ui)( Σk=0υk) ≤ ( Σi=0ui)( Σk=0υk), n = 0,1, … .

(3)

Having an upper bound, the sequence Cn has a least one (due to the corresponding, or any, axiom of completeness (4.3)); since Cn increases it has a limit. – – The Cauchy product of absolutely convergent series converges absolutely. This, through

Σnl=0 |Σli=0 ui υl – i | ≤ Σnl=0 Σli=0 | ui | | υl – i | , n = 0,1, … , is clear by the Proposition and the arguments employed there. * * * Theorem 3.3 (Cauchy). ∞

{1b}

Let ui ≥ 0, υk ≥ 0;

∞

∞

l

if Σi=0 ui = U, Σk=0 υk = V then Σl=0 Σi=0 ui υl – i = UV.

(4)

Proof. According as mn = (n – 1)/2 or mn = (n – 2)/2 is an integer, the terms of the sequence Cn in (3) get squeezed in by n

n

(Σ0 ≤ i ≤ m ui)(Σ0 ≤ k ≤ m υk) ≤ Cn ≤ (Σi=0 ui)(Σk=0 υk), n

n

showing lim Cn = UV as n → ∞. – – For the following proof, note the ∞

∞

Remark. Let 0 ≕ n0 < n1 < n2 < … . If (S) Σk=0 ai converges, so does (S*) a0 + Σk=1

Σnk–1 < i ≤ nk ai (to the same value) since the partial sums of S* are partial sums of S. Cauchy generalizes Thm. 3.3 (not being followed in {6}) by the next one, which proved to be an important tool, namely when applied to power series. ∞

∞

Theorem 3.4 (Cauchy).{1c, 2a, 3b, 4b} Let Σk=0 ui = U, Σk=0 υk = V; assume both series to converge absolutely. Then their Cauchy product absolutely converges to UV. Proof. Cauchy lines up the terms uiυk (i, k = 0,1, ...) two ways, each starting from u0υ0. The first route goes along two sides each of the subsequent squares (0,1, ... , n)×(0,1, ... , n), n = 1,2, ... , namely thus: run the low line (n,0), (n,1), ... , (n,n), then run the right hand column (n –1,n), (n –2,n), ... , (0,n). Let this course be described by the one-to-one mapping ν ↦ (iν , kν ) of ℕ0 onto ℕ0 × ℕ0 and let S denote the series with the partial sums n

sn = Σν=0 uiνυkν.

3.4 Cauchy

63

By assumption, Cauchy gets n

n

s(n+1)2–1 = Σi=0 Σk=0 u i υk =

(Σni=0 u i )(Σnk=0 υk ) → UV

as n → ∞.

(5)

Hence, there exists a subsequence of sn which converges to UV. In order to verify sn → UV, we are left to prove the convergence of (sn). When running the matrix ( | uiυk | : i, k = 0,1, ...) the same route, we likewise get an increasing sequence –s ≔ n

Σnν=0 | uiνυkν|

which is bounded because of n

n

s–(n+1)2–1 = (Σi=0 | u i |)(Σk=0 | υk |) ≤

∞ (Σ∞ | u |)(Σk=0 | υk |) , i=0 i

n = 0,1, … .

This shows

Σ| ui υk | < ∞ ν

(6)

ν

and, in particular, the series S to converge which, together with (5), yields sn → UV as n → ∞.

(7)

Now, let a sequence (pν, qν) follow successively the slanting lines (cf. (2)) that form the ∞

Cauchy product; let S* denote the series Σν=0 upνυqν. If we know that its partial sums sn* converge to UV, the Remark yields what we want. In view of (7), it suffices to prove sn – sn* → 0 (cf. {2b}). Let us write wν ≔ uiνυkν and (pν, qν) ≕ (iμ(ν), kμ(ν)) with a one-to-one mapping ν ↦ μ(ν), ν∈ℕ0. [Thus, the term wν in series S becomes the term wμ(ν)* in series S*.] By virtue of (6) we know: Given ε > 0 there is nε such that Σnε < ν ≤ nε + h | wν | ≤ ε holds whatever be h = 1,2, ... . With Nε sufficiently large we get {0, 1, ... , nε} ⊆ {μ(0), μ(1), ... , μ(n) } for all n ≥ Nε. Colloquially: Among the terms in sn*, all terms w0, w1 , ... , wnε occur. Therefore, in the aggregate n

n

sn – sn* = Σν=0 wν – Σν=0 wμ(ν)* all terms w0 , w1 , ... , wnε cancel out, so that the difference only comprises terms ± wν where ν ≥ nε. Hence we arrive at | sn – sn*| ≤

Σn < ν ≤ n + h | τν | ≤ ε,

with some hn , for all n ≥ Nε . – – * * *

For a moment, let us question the absolute convergence as a hypothesis in Thm. 3.4. (Cauchy does not say “absolute”; he takes “the numerical value” of real terms.) The condition imposed in {1c} beyond convergence of the factor series accounts for the sequence sn to get rearranged into the sequence sn* with the same limit. It thus proves indispensable in that proof.

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3 Series of functions: the prototypes

Cauchy gave evidence that the Theorem cannot go unpunished if it drops “absolute”. When forming

(Σ∞ i=0

(–1)i (–1)k ∞ )(Σ )= k=0 k + 1 i+1

Σ∞ Σl l=0 i=0

(–1)i i+1

(–1)l–i {1d} l–i+1

(8)

with a series on the left that converges by Leibniz’ test{2c}, the “Cauchy square” diverges since

| (–1)l Σli=0

1 (i + 1)(l – i + 1)

|≥

l+1 = 1.{2d} (l + 1)(l – 0 + 1)

In fact it is not necessary for any one of the factor series to converge at all in order that the Cauchy product becomes even absolutely convergent, this being substantiated (see Appx. C-14) by ∞ Σ∞ u = 3 + 31 + 32 + … , Σk=0 υk = – 2 + 21 + 22 + … . i=0 i

(9)

We will revert to this in 4.2.3 when discussing the words “necessary condition” as a means of mathematical communication. * * * When the problem reduces to ask for the ordinary convergence of the Cauchy product, a first and easy result is due to the Austrian Franz Mertens. About fifty years after Cauchy’s publication he proved ∞

∞

Theorem 3.5 (Mertens).{2e, 3c, 5d} If Σi=0 ui = U, Σk=0 υk = V and if one of the series converges absolutely, then their Cauchy product converges to UV. For the proof see Appx. C-15. In 1908, G.H. Hardy proved (4) to hold for all convergent series Σun, Συn where nun, nυn form bounded sequences.{2f} For earlier results see 5.2.3: Abel furnished a proof for (4) with but ordinary convergence at the expense of a condition on the Cauchy product itself, Ernesto Cesàro would do so in another way. * * * All the information in Cauchy’s Thm. 3.4 is covered when Mertens’ Thm. 3.5 is joined by the Proposition, or Cauchy’s Thm. 3.3. The access to Thm. 3.4 via Mertens’ Theorem is more appropriate and, furthermore, much simpler. Nevertheless, Thm. 3.4 is most noteworthy due to its proof. It became a landmark in analysis because its author discovered that rearrangement of real terms in a non-absolutely convergent series might change its value. (This is reported in {2g}; Cauchy [5] is not at my disposal.) Bernhard Riemann was the first to fully disclose how rearrangement of real series relates to absolute convergence.{2h} {1}

CAUCHY [2]: {1a} 127 bottom. {1b} 127 f, § II/Théorème VI. {1c} 132–134, § III/Théorème VI. {1d} 134 f.

3.4 Cauchy {2}

{3} {4} {5} {6}

65

KNOPP: {2a} [1] 146 f, misprint 146 foot n.2: read 132; [2] no.91; [3] 91. {2b} [1] 140; [2] no.88; [3] 79. {2c} [1] 132; [2] no.81c; [3] 68 f. {2d} [1] 148 f; [2] no.91; [3] 91. {2e} [1] 330 f; [2] no.81c; [3] 145 f. {2f} [1] 332 footn.1; [2] no.189, 2nd footn.1; [3] 147 f. {2g} [1] 139, footn.1; [2] no.87, 2nd footn.1. {2h} [1] 328 ff; [2] no.187. HARDY: {3a} 227. {3b} 228, Thm.160. {3c} 228, Thm.161. BOOS: {4a} 31. {4b} 28 ff. RUDIN: {5a} [1] 63; [2] 82. {5b} [1] 66 ff; [2] 85 ff. {5c} [1] 64 f (65 line 1?); [2] 83 (line 15?). {5d} [1] 65 f; [2] 83 f. {5e} [2] 86 f. REIFF 169 f.

4

Series seriously: the Greek comet reappears

4.1

Rigor in retrospect and prospect

Our history of Western analysis splits into the archaic period and the one which, for one thing, culminated in the era of Newton, Leibniz, Euler. In between, there was no Western tradition. So it is little wonder that the epochs share the pattern of development, namely productive speculation and methodical consolidation. Leaving behind any speculation, the Greeks eventually excelled in rigor, personified by Eudoxus and Archimedes. They handled the infinite the indirect way. On this ground calculus could hardly have developed and flourished. Its protagonists were facing the infinite directly, on the grounds of pragmatism. Newton and Leibniz did not have the Greek drive to perfection and took chances with harsh criticism. Given the gold rush to exploit what had become accessible then, why should their followers feel particularly invited towards fundamental thought? Aside of rigor, excellent mathematics was accomplished. The end had to sanctify the means… Few analysts were missing the spirit of the ancients, many were sharing the attitude which, in 1810, made the author of a popular French textbook say:“Those sophistries that bothered the old Greeks, we don’t need them any longer.” {1a} The great Carl Gustav Jacob Jacobi, by the time of 1840, felt that “for… rigor we have no time”.{1b} Lagrange was a lonely voice with hardly an echo (3.2.4 end). In the basement of analysis there was little light at the Age of Enlightenment. * * * How is “convergence” to be given a meaning, how without the infinitely small? It was this virus that resisted radical curing. Cauchy’s notion of convergence pertains to “a variable that is successively assigned values to approach a fixed value without end” (“les valeurs ... s’approchent indéfiniment d’une valeur fixe”){2a}. As to specify the procedure, Cauchy continues: “ ... , de manière à finir en différer aussi peu que l’on voudra …” [such that they eventually differ from it as little as ever one wants]. This in fact is the germ of convergence, yet no much more than some loose description, not a prescription how to measure approach. (The latter is a must. Like in physics where a quantity is not defined unless a prescription tells us how to measure it.) And, after all, one must ask how to fix“une valeur”. The crucial issue is a twofold one, comprising the process and its goal. “Converging” must be freed from the connotation of a move and its goal must be provided mathematical definition. The foremost goal in history was π. According to Plato a well defined quantity. Does it qualify to be what we call a number? To be or not to be – the question became recognized and answered when the epoch ended.

68

4 Series seriously: the Greek comet reappears

As long as “converge” remained synonymous with “converge to a number” substance 1 was lacking in many a case. The symbol lim n looks self-evident, it is not. How much less 1

∞

1

might lim (1 + n )n, ∑k=0 k! be entitled to play the role of a number? What in the world is it that a sequence converges to in case it is not a rational number? For ages, there was a hole to be filled “provisionally” by a symbol, a dummy. Yet, there is more at stake than a quantity to be squeezed in between rational numbers. It must irreproachably fit the rules of calculation! To create such kind of numbers preferably fell to infinite sequences and series. * * * We today are used to knowing: A series that complies with the Cauchy criterion owns a sum, an increasing sequence that has an upper bound owns a least one. On account of this, conclusions like the following could not be thought out in the past. If a series converges absolutely, then it converges. (Mind the language! Cf. Appx. C-4.) If a sequence (bk) is bounded then Σ bk xk converges for x ∈( –1, 1). (A decimal fraction, for example.) It only looks like Cauchy characterizing convergence of a series Σuk without recourse to a prospective limit number when he conceived a property which, in our terms, reads as follows: To any ε > 0, there is a number Nε such that |un +…+ um| ≤ ε holds whenever m ≥ n ≥ Nε. Cauchy maintains to have proved it necessary and sufficient in order that the series is convergent “to some number”. In fact, he but pays regard to the necessity part which was stated by Euler{3} already. As to the very point, Cauchy just notes: “Réciproquement, lorsque ces diverses conditions sont remplies, la convergence de la série est assurée.”{2b} Assured in what sense? He trusted in geometry. Unlike Bernhard Bolzano from Prague who proved sensitive as to the principles of analysis{4}, not only when trying to ground the irrational numbers on the rationals. It is a must to acknowledge the work of that notable pioneer of analysis (cf. {4b,c}). Series that fulfill Euler’s and Cauchy’s postulate are called Cauchy series. Their partial sums, the Cauchy sequences, also go by the name fundamental sequences due to the fundamental part they play in analysis (4.3, 4.4, 5.3.2). It has become standard to denote Cauchy’s statement the Cauchy criterion, as comprising a necessary and sufficient condition on convergence. Quite some writers make suggest that it was proven by Cauchy in full.{5} Both the aspects of the limit, i.e. definition and value, join in a program that will be called the arithmetization of analysis{6}. It means to found analysis on numbers alone: for one thing on the logical relations between number relations and, basically, on the very concept of number, without substantial resort to geometry. This final goal considered to be attained, is it to become final possession? Arithmetization has its price, it only can be achieved at the expense of the infinite. Not the kind which makes series infinite. It is the infinite sets. Analysis operates on those sets, its foundation cannot dispense with them. And set theory would soon fall into disrepute… In historical order, foundations would take last place (4.3). Here, we will follow the revision process up to 1900. To start with in 4.2, we focus on how the mode of analytical reasoning developed.

4.2 Farewell to the infinitely small; the “epsilontics” {1} {2} {3} {4} {5} {6}

69

{1a} HEUSER [2] 689 top. {1b} KLINE [2] 166. KÖRLE 70. CAUCHY [3]: {2a} 19. {2b} 116. REIFF 118 ff. GRATTAN-GUINNESS 52 ff. HEUSER [2] 689 f. LÜTZEN 219–222. KÖRLE 74. SONAR 500–502 (read “Zwischenwertsatz” on 500, 703). BOYER 566. ANGLIN 200. CAJORI 398–400. HOBSON [1] 22 f. EDWARDS, JR. 333. BOTTAZZINI 327.

4.2

Farewell to the infinitely small; the “epsilontics”

4.2.1

Convergence, continuity

James Gregory, a contemporary of Newton and Leibniz, who no less knew about calculus coined the word “convergence”, applying the idea to solve Zeno’s puzzle on Achilles and the tortoise (1.1, 1.3). He who writes 1/n → 0 and talks accordingly, suggests a move in space and time which at best is a metaphor in place of a mathematics’ predicate. Cauchy talked approach all the way. Convergence requires appropriate “words and syntax”. And it required some people to take care of that. They had to translate an immaterial move into proper language, say arithmetical and logical relations. Indeed, there was someone to fully fix the syntax. To fix – in its literal meaning: by a definition of limit that is static. Karl Weierstraß, a former boozy student, would shape up analysis while teaching the ABC in a one-class school. When he eventually held a professorship Weierstraß performed quite like Cauchy to whom teaching meant professing his subject. However, he was the one who would make precise what Cauchy had touched upon when introducing, yet not going through with, the epsilon. Weierstraß says: [1/n → 0] is shorthand for [given ε > 0 there is some number Nε such that 1/n ≤ ε holds whenever n ≥ Nε]. (For tautology, choose Nε = 1/ε here.) * * * Euler’s continuous functions were “expressions” (3.2.6), in fact differentiable functions{1}. Cauchy refers continuity of a function to a “variable” with values “intermédiaire entre deux limites données”{2a} and gives a verbal description in the language of “infinitely small increments”{3}:“un accroissement infiniment petit de la variable produit toujours un accroissement infiniment petit de la fonction elle-même”{2a}. (Cauchy wants the idioms“un infiniment petit“,“une quantité infiniment petite”{2b} to be interpreted as a “variable whose values decrease below any number”; the way they are referred to makes them smell no other than the infinitely small in past speculations.) We might think of functions on domains D ⊆ ℝ that consist of limit points (“Häufungspunkte”{4a}) ξ in that every neighborhood of ξ contains some u ∈D with u ≠ ξ, even though{4b} any non-empty set may form a domain to conceive continuity at a point equivalently by

70

4 Series seriously: the Greek comet reappears

Definition 4.1. Let f : D → ℝ, ξ ∈D ⊆ ℝ; f is continuous at ξ if ( i) f (xn) → f (ξ ) whenever D ∍ xn → ξ ; (ii) given any ε > 0, then there is some δε[ξ]> 0 such that | f (x) – f (ξ ) | ≤ ε whenever | x – ξ | ≤ δε[ξ], x∈D. The function is called continuous on D0 ⊆ D if it is continuous at every ξ ∈D0 and plainly called continuous in case that D0 = D. (For distinction, “pointwise continuous” and “global continuity” are in use.) Euler would have been much astonished to hear a function called continuous when defined at finitely many points only. (You too? Do Appx. D-1.) Each of the versions (i), (ii) in Def. 4.1 has its place. Intuitively, limits are “approached”, stepwise. An image that is being preserved and legalized through (i). (Infinity beyond countability is involved at (i), too, since this so-called sequence criterion of continuity refers to all sequences (xn) specified.) In case, students might be recommended to employ it.{5} * * * People want to know what is true; no less, in case more importantly, is to know what is false. For a conjecture in classical mathematics one either asks for a proof, or a disproof like a counterexample. Let us consider the clause “f (x) is continuous at x = ξ ”. Given Cauchy’s “definition”, how should a statement be verified or falsified? Here we go with respect to both parts of Def. 4.1. For “not i” (“non i” in Latin) we have to show: There exist xn ∈D such that xn → ξ and f (xn) ↛ f (ξ ). We put the two statements f (xn) → f (ξ ), f (xn) ↛ f (ξ ) over one another: [For any ε > 0] { [There exists ε > 0] there holds such that

such that

[there exists N ] such that [for any N]

| fn (x) – f (ξ ) | ≤ ε; | fn (x) – f (ξ ) | > ε.

[for any n ≥ N] [there exists n ≥ N]

}

The transition from one line to the other is mechanically performed when the quantifiers [there exists], [for any] are switched from line to line. They are named the existence- and the all-quantifier (however note that one clearly should not say “for all ε > 0 there exists some N ”!) And this is how to negate “ii”: [There exists ε > 0] such that [for any δ > 0] [there exists x ∈(ξ – δ , ξ + δ) ∩ D] such that | f (x) – f (ξ ) | > ε. * * * Giving credit to Weierstraß for arithmetization must not belittle what we owe to so many. Think for instance of a Bolzano-Weierstraß theorem, originating as early as 1817{6}. Bernhard Riemann made it a strong point that the notion of limit and the like has to be grounded on factual distances and their relations (confer {7a} in his doctoral thesis, 1851, and, explicitly, at {7b}).

4.2 Farewell to the infinitely small; the “epsilontics” {1} {2} {3} {4} {5} {6} {7}

71

GRATTAN-GUINNESS 7 top. CAUCHY [3]: {2a} 43. {2b} 19. EDWARDS, JR. 310 f. RUDIN: {4a} [1] 28; [2] 35. {4b} [1] 74; [2] 98. KÖRLE 75 f. KNOPP [1] 92, footn.1; [2] no.54. RIEMANN: {7a} [1] 3, footn.(1) on 46 top. {7b} [2] 111 (“7.Nov.1855”).

4.2.2

Dirichlet, Heine: uniform continuity

Newton considered integration as being luckily reduced down to anti-differentiation, and it meant a great accomplishment when the fathers of Calculus provided easy access to the values of integrals they encountered. But it was progress at the expense of another value: the integral’s heritage from rooting in geometry. Cauchy was determined to re-implant the definite integral when he re-defined it through refining lower and upper sums of rectangles as Blaise Pascal{1, 2a, 3a} once did. Instead of “figuring out the integral”, Cauchy asked for integrability, and in fact the criterion{2b, 4b} he conceived within {5} matches Riemann’s integral. He was sure of having proved: If a positive real function f is continuous on [a, b] then b

∫a f (x) dx exists. Aside from the issue of number existence (cf. 4.1, 4.3), Cauchy’s proof is invalid, the reason being: He was not able to make the most of his hypothesis in order to satisfy his criterion. Actually, Cauchy exploited continuity but on [a, b); definitively, his argument would 1

prove ∫0 1/(1– x) dx to exist.{2c} Dirichlet and Riemann put right what Cauchy had missed. From continuity of f on a closed bounded interval they deduced a property of f which significantly exceeds global continuity. Here it is. In Def. 4.1(ii), the tolerance of | x – ξ | was denoted δε[ξ] to indicate that it possibly depends on the point ξ of continuity. If, given ε, we want to say that (ii) holds for some δε irrespective of ξ ∈D, we may do so by the following Definition 4.2. Let f : D → ℝ, then f is called uniformly continuous (on D) if, given ε > 0, there is some δε > 0 such that | f (u) – f (υ)| ≤ ε whenever | u – υ | ≤ δε , u, υ ∈D. Verification and application mostly pertain to “compact” domains D like [a, b]. (Do Appx. D-2.) Cauchy’s argumentation would have been substantiated by virtue of Theorem 4.1. A continuous function [a, b] → ℝ is uniformly continuous. A truly fundamental and far-reaching result in analysis! It was published by Eduard Heine{6}, first in 1870, and apparently had been found by Dirichlet no later than 1854. Cauchy’s definition of continuity refers to a variable x “entre deux limites” (cf. 4.2.1), but in no way it covers the interpretation needed. Its wording does not allow for realizing the defect in his proof. There are those who deny any, deliberately or not (“Cauchy could prove …” {3b, 7}). In two compendiums, the said paper happens to be specifically celebrated: “On this occasion, an important new notion will make its appearance, that of uniform continuity.”{4b} [translated], and, explicitly: “Cauchy utilise implicitement la continuité uniforme”{4c}. In {3c} we read:

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4 Series seriously: the Greek comet reappears

The way Cauchy uses the notion of continuity seems to make uniform continuity “selfevident”. The proof does not display any trace of this. EDWARDS, JR. 112 f. KÖRLE: {2a} 147–150. {2b} 195. {2c} 196 f. LÜTZEN: {3a} 212 middle (but cf. {1},{2a}, as to Fourier). {3b} 212 bottom. {3c} 207 middle, 214. DUGAC: {4a} 354 f. {4b} 354 upper. {4c} 354 bottom. CAUCHY [4] 122–125. HEINE 361. HOCHKIRCHEN 331 top.

{1} {2} {3} {4} {5} {6} {7}

4.2.3

From Abel to Weierstraß: uniform convergence

We are heading for a significant milestone in the history of function series. Abel’s inquiry into (real) power series convergence disclosed an important feature. It has something in common with uniform continuity. The function 1/(1 – x), being pointwise continuous on (0,1), is even uniformly continuous ∞

on (0,½], yet not on (0,1) (why not? Cf. Appx. D-2). The series Σk=0 xk is pointwise convergent on (0,1); as concerns (0,½] we will encounter convergence of some higher standard. * * * A power series which converges on an open interval constitutes a continuous function there ∞

(see the report in 3.2.4). If a series Σk=0 ak xk converges at some x1 > 0 it also converges on (0, x1) (see 3.2.4 (4)) and generates a continuous function thereon. Is it continuous on (0, x1] also in the case that x1 is a boundary point of the (open) interval of convergence? Let us consider some functions that have Taylor expansions about x0 = 0, with r = 1 the radius of convergence. What may happen at x = x1 = 1? There, the geometric series does not converge nor does its sum 1/(1 – x) converge as x → 1– 0. The Taylor series of 1/(1 + x) diverges at x1 = 1 whereas the function is continuous at x = 1. Now, consider both the Taylor series of ln (1 + x) and arctan x about x0 = 0. They converge at x1 = 1 as shows the Leibniz test (cf. 3.2.2, 3.2.1). Mengoli (3.2.2) and, respectively, Nīlakantha and Leibniz (3.2.1) took it for granted that, with x ∈(0,1), the following is true: ln (1 + x) = arctan x =

(–1) Σ∞ k=0 k

k

xk →

k

(–1) Σ∞ k=0 k

k

, as x → 1 – 0 , k

(–1) 2k + 1 ∞ (–1) Σ∞ x → Σk=0 2k + 1 , k=0 2k + 1

as x → 1 – 0 .

These are Taylor expansions on (0,1). They extend to (0,1] which is made explicit through n

(–1)k

n

(–1)k

lim x → 1–0 lim n → ∞Σk=0 k x k = lim n → ∞ lim x → 1–0 Σk=0 k

xk

4.2 Farewell to the infinitely small; the “epsilontics”

73

and the like. The proof thereof had to wait for Abel. He apparently was the first to realize that those “changing limits” need to be verified by proof. * * * Abel’s famous Limit Theorem{1a, 2a, 3} of 1826 asserts Theorem 4.2. If Σ ak x k converges for x = 1 then

Σ∞ a xk k=0 k

→

Σ∞ a k=0 k

as x → 1 – 0 .

(1) n

Proof. For convergence of Σ ak xk on (0,1) see 3.2.4(4). – We suppose that An ≔ Σk=0ak → 0 and leave the rest to Appx. D-3. With A–1 ≔ 0, we note

Σnk=0 ak x k = [Σnk=0 (Ak – Ak–1) x k

=

n–1 Σnk=0 Ak x k – Σk=0 Ak x k+1 =]

n–1 Ak (x k – x k+1) + An xn . Σk=0

(2)

As (An) is bounded, Σ Ak x k converges absolutely on (0,1); the sequence (2) has for limit ∞

∞

A(x) ≔ Σk=0 ak x k = (1 – x)Σk=0 Ak x k, 0 < x < 1. ε

Given ε > 0, choose k(ε ) ∈ℕ such that |Ak | ≤ 2 for all k > k(ε ). Hence, with M(ε) ≔ k(ε ) Σk=0 | Ak | there holds

ε

1

| A(x) | ≤ (1 – x)M(ε) + (1 – x)Σk > k(ε ) | Ak | xk ≤ (1 – x)M(ε) + (1 – x) 2 1 – x ≤ ε ε

for all x ∈ (0,1) such close to 1 as to make (1 – x) M(ε) ≤ 2 . – – Thm. 4.2 applies to the examples ahead and also to the arcustangent series at x = –1 (why?). – It readily is adapted to the case where Σ ak x 1 k converges for some x1 > 0 in general (cf. 3.2.4(4) and its sequel): If Σ(ak x 1 k ) ξ k is assumed to converge for ξ = 1 then ∞ ∞ Σk=0 ak x k = Σk=0 (ak x 1 k ) ξ k

→

Σ∞k=0 ak x 1 k

for x = x 1 ξ → x 1 as ξ → 1 – 0.

* * * Abel’s limit theorem was the first step to disclose a particular kind of convergence to occur with series and sequences of functions. As it results in the change of limits, Abel would surely have analyzed it thoroughly if he had not died shortly after his paper appeared. More than a decade later interest was rising to describe the difference by which, say, the two series Σ(–1)k xk, (–1)k Σ k xk converge on intervals (1– δ, 1). Several people engaged themselves until Christof Gudermann and then his student Weierstraß{4a} got to the very bottom. (As to Cauchy, see below.)

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4 Series seriously: the Greek comet reappears

Their insight is fixed in the Proposition below. To prove it, we look to the proof of Thm. 4.2, supposing An → A with A = 0 in the first place, and apply Cauchy’s criterion (in 4.1) to k k+1 n l Σnk=l+1ak xk = [Σnk=l+1(Ak – Ak–1) xk =] Σn–1 k=l Ak (x – x ) + An x – Al x ,

l ≥ –1,

(3)

where the case l = –1 with A–1 ≔ 0 is (2) above. Given ε > 0, there is Nε such that k |Σnk=l+1ak xk | ≤ ε3 (1 – x)|Σn–1 k=l x | +

ε

ε

3+3 ≤ ε

for all n ≥ l ≥ Nε and all x ∈(0,1] (!). This proves the Proposition in case of A = 0. Now, let An → A in general. The right hand equation at (3) being an identity, we can make the null ∞

sequence of remainders Rk ≔ Σj=k+1 aj x j = A – Ak play the role of the null sequence An in (3) and thus get n

– Σk=l+1 ak x k =

Σnk=l+1 (Rk – Rk–1) x k

=

k k+1 n l Σn–1 k=l Rk (x – x ) + Rn x – Rl x ,

l ≥ –1.

When applied again, the previous argument assures the ∞

Proposition. Let Σak converge, A(x) ≔ Σk=0 ak x k , 0 < x ≤ 1. Then, given ε > 0, there is Nε such that ( i)

| Σnk=0 ak x k

(ii)

| Σnk=0 ak x k –

–

l Σk=0 ak x k | ≤ ε

holds for all n, l ≥ Nε and all x ∈(0,1];

A(x)| ≤ ε holds for all n ≥ Nε and all x ∈(0,1],

where the last line follows from the preceding by l → ∞. Like it was done with Thm. 4.2, the Proposition can be extended to yield: Let Σ ak x1k converge for some x1 > 0 then, given ε > 0, there is Nε such that (i), (ii) hold with “x ∈(0,1]” replaced by “x ∈(0, x1]” . Remarks. 1. Equations (2), (3), with […] being dropped, remind one of partial integration in calculus. “Theorem III” in {1b} covers the construct which came to be known as Abel’s Partial Summation{2b}. 2. Re: Proposition line (i). If, with all n, l ≥ Nε, the inequality holds for all x ∈(0,1) then it also holds for all x ∈(0,1] since the functions x ↦ xk are continuous at x = 1. * * * In the sequel, we will expound the idea of the Proposition. The latter displays a property of a series Σ ak x k the convergence of which is due to convergence of Σak. However, we will no longer confine ourselves to the behavior of power series. Cauchy preferably thought in terms of series rather than sequences, but seemingly he did not think of ∞

1+ Σk=1 (xk – xk–1) , 0 < x ≤ 1,

(4)

4.2 Farewell to the infinitely small; the “epsilontics”

75

which, not a power series, generates the “power sequence” of partial sums fn(x) = x0, x1, x2, … , 0 < x ≤ 1.

(5)

Have a look at Fig. 4.1 with the graphs of f0, f1, … , f5. The sequence (5) of continuous functions converges on [0, 1], namely by limn fn (x) ≡ 0 (0 ≤ x < 1), lim fn (1) = 1. A discontinuous limit function indeed! (On (0, 1) the limit is continuous.) As to Cauchy, see more below. Other than Σ(–1)k xk, the series (4) converges at x = 1. As regards the mode of convergence on (0, 1), we will see both these series to behave the same – and different from the logarithmic or the arcustangent series. How does the convergence of the sequence (fn) on (0, u] with u < 1 compare to its convergence on the domains (0, 1] or (0, 1)? Given ε > 0, there exists nε such that 0 < fn (x) ≤ fn (u) ≤ ε for all n ≥ nε and with 0 < x ≤ u

(in Fig. 4.1, we chose nε = 3). On the other hand, provided 0 < ε < 1, to however great an N there always is some υ ∈(0, 1) such that fN (υ) > ε. Symptoms like these gave rise to what Weierstraß, once upon a time, put on a little blackboard, eagerly grasped by a big audience. It is the spirit of x0, x1, x2, x3, x4, x5

1

ε 0

u

υ 1x

Fig. 4.1. Uniform convergence of (5) on any (0, u), u < 1; non-uniform convergence of (5) on (0, 1).

Definition 4.3.{2c} A sequence fn : D → ℝ is said to converge to f (x) ≡ 0 uniformly on D ⊆ ℝ if, for every ε > 0, there is some nε such that | fn(x) | ≤ ε holds for all n ≥ nε and any x ∈D. Uniform convergence gn → g is defined by uniform convergence of fn ≔ gn – g to f (x) ≡ 0. Def. 4.3 applies to a series of functions through the sequence of its partial sums. – If the functions fn are continuous on (a, b], then uniform convergence of the sequence on (a, b) infers its uniform convergence on (a, b]; see Remark 2. (Do Appx. D-4, D-5.) Through Def. 4.3 the Proposition above when extended accounts for Theorem 4.3. If Σ ak x 1 k converges then Σ ak x k converges uniformly on (0, x 1 ]. * * *

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4 Series seriously: the Greek comet reappears

We started out from Abel’s limit theorem. Its proof made us discover uniform convergence. Would Thm. 4.3 lead us back to Abel’s theorem? Yes, the guide being Theorem 4.4. Let fn : D → ℝ be continuous. If fn → f uniformly on D then f is continuous on D. Proof. In view of Def. 4.1(ii) we consider ξ ∈D fixed. Given ε > 0 then, by uniform converε

gence, there exists nε ∈ℕ such that | f (x) − fnε(x)| ≤ 3 for all x ∈D. Hence, by continuity of fnε at x = ξ, we get | f (x) − f (ξ) | ≤ | f (x) − fnε(x) | + | fnε(x) − fnε(ξ) | + | fnε(ξ) − f (ξ) | ε

ε

≤ 3 + | fnε(x) − fnε(ξ) | + 3 ≤ ε for all x ∈D sufficiently close to ξ. – – (Note that Def. 4.3 is employed with n = nε only.) Almost everything on real power series in 3.2.4 had been compiled without proof. Regarding continuity, access is now provided by the following Summary of Thms. 4.3, 4.4. If the series Σ ak x k converges for some x = x1 > 0, then it con∞

verges uniformly on (0, x1] and the function x ↦ Σk=0 ak x k is continuous on (0, x1]. This essentially covers that a power series is continuous on its interval of convergence and it comprises{2d} Abel’s result on boundary points of the interval. * * * Uniform convergence plays a central role in analysis. It provides a sufficient condition for the traditional term-by-term technique “in ℝ” as concerns limits and integration (yet not differentiation). G.H. Hardy esteems Weierstraß to be the one who not only gave the clearcut definition of uniform convergence, but also fully recognized its significance.{4b} Unfortunately, there is plenty of, say, misunderstanding. In black and white. What, for instance, does Thm. 4.4 tell us about the sequence (5)? If it would converge on (0, 1] uniformly then its limit function would be continuous on (0, 1]. It is not, it jumps. Consequently, (5) does not converge uniformly on (0, 1]. But what about (0, 1) in its place? If we assume (5) to converge uniformly on (0, 1) then the limit function would be continuous on (0, 1) – but it is! No contradiction, no decision; here, Thm. 4.4 is inconclusive. (The assumption is easily refuted directly; do Appx. D-4!) We will employ the sequence (5) to demonstrate a paradigm of fallacy. Let any continuous functions φn converge to φ on D ⊆ ℝ. How do then the following statements relate?

φ is not continuous on D; (φn ) does not converge uniformly on D.

(A) (B)

Thm. 4.4 equivalently says “ if A then B ”, i.e. the indirect way (because in case of uniform convergence we would have continuity, contrary to A).

4.2 Farewell to the infinitely small; the “epsilontics”

77

When reading A, B in this order, somebody might be tempted to put “no wonder, since ...” in between. Indeed, there are not just a few somebodies who more{5a} or less{5b} explicitly make printed that the limit function could not be continuous unless the convergence is uniform. In other words: uniform convergence would be a necessary condition in order that the limit be continuous. It definitively is not, B does not infer A: The sequence φn(x) = xn, x∈D = (0, 1), has a continuous limit, yet does not converge uniformly on D since otherwise its extension to (0, 1] would also converge uniformly (see Remark 2) and thus have a continuous limit. (For a historical counterexample close to Georg Cantor’s{6} see Appx. D-6{7a}.) The same trap works with term-by-term integration.{8} (Cf. {7b} for both continuity and integration.) What might account for such a mess, unworthy of a mathematics discourse? Within a specific proof where uniform convergence is employed one may be unable to do without; there it might actually be needed{9a}. Necessity of a predicate is something else.{9b} (See 3.4.2(8) for a lesson on this.) And beware of “the condition”! Of a wording like “uniform convergence as condition for ...”{9b}! Colloquial language employs “conditions” in the sense of circumstances; in 3.3.3 we met boundary and initial conditions. The bare singleton “condition”, neither specified by attribute nor by context, becomes open to contrasting interpretation. * * * It does not have to do with a necessary condition that Thm. 4.4 becomes false if “uniformly” is dropped − an erroneous result which Cauchy published in 1921{10a}. (See {4a} for how the story started.) Abel respectfully put forward his criticism: In the paper where his Limit Theorem was published we find a footnote{1c} with reference to the sawtooth function of 3.3.2(3): “... it seems to me that this theorem allows for exceptions.” Cauchy republished{4b} his result after Abel’s death and kept to it until 1853{4c} when he released a correction{4d, 10b}. This made Bourbaki le général say Cauchy had “believed for a moment that a convergent series of continuous functions has a continuous function for sum” (“il avait cru un moment que ...” ){11}. (It is noted at {4b} that Gudermann’s concept of uniform convergence dated 1838 and that it was Cauchy who, by publication in 1853{4c}, introduced the idea“pour la premiere fois”, for the first time.) Our freshwomen and -men may take comfort from the troubles (cf. 4.2.2) which such a great mathematician had been stricken with. Weierstraß made uniform convergence the instrument for his own theory of functions. Cauchy’s was based on integrals (3.4.1); now, series proved to be equally efficient. Newton, Leibniz, Euler, Cauchy having played kick and hope with infinite series, luckily compare to a soccer team in that their offside goals subsequently were confirmed by uniform convergence. It was to open a spate of consequences in more and most advanced calculus. {1} {2} {3} {4}

ABEL: {1a} [1] 314 f, Lehrsatz IV; [2] 223, Théorème IV. {1b} [1] 314, Lehrsatz III; [2] 222 f, Théorème III. {1c} [1] 316 footn.; [2] 224 f, footn. KNOPP: {2a} [1] 179 f; [2] 177; [3] 109 f. {2b} [1] 322 f; [2] no.182. {2c} [1] 342; [2] no.191. {2d} [1] 359, Footn.2; [2] no.199(2.). RUDIN [1] 160; [2] 203 f. DUGAC: {4a} 349–352. {4b} 351 f. {4c} 352 f. {4d } 353.

78

4 Series seriously: the Greek comet reappears

{5}

{5a} CAJORI 377, lines 9, 10. KLINE 965 middle. LÜTZEN 234. {5b} HEUSER [2] 697 („entscheidend“, i.e., crucial). {6} CANTOR 267. {7} KÖRLE: {7a} 210(3). {7b} 77, 212. {8} CAJORI 376, 377 line 1. KLINE 964. LÜTZEN 237. {9} HOCHKIRCHEN: {9a} 345 below (quotation). {9b} 346 lines 4, 5. {10} CAUCHY: {10a} [3] 120. {10b} [2] 31 f. {11} BOURBAKI [2] 150 lines 9–7 from below.

4.3

Welcome to irrationals: the complete space of real numbers

Let us look back to when and where the history of real numbers began, to the story about how irrational numbers were detected as disastrous specters. It is worth a close-up to Pythagoras announcing “all is number” and witness the first loss of certainty in mathematics. The master had a toy named monochord, a one-string guitar which he played in shortening the string by ratios like 1:8, 2:5. It answered through harmonics, accords. Pythagoras held all world to be ruled ”accordingly“. The world, to him it embodied geometry. So, harmony had to be inherent in all geometry. In academic Greece numbers meant line segments. Given any such A, B, the Pythagoreans were sure about numbers m, n = 1,2, … to exist in order to fulfill the proportion which we write A:B = m:n (Fig. 4.2 shows A:B = 3:4) and which in Greek reads “n times A makes m times B”. These people could (and would?) take parts like A/m, depicted in Fig. 4.2 where the segments A, U are measured along rays, off from their vertex and marked by their endpoints A, U (there, “mU” for example is to stand for the endpoint of the segment mU). This will help us to interpret and visualize n A = m B. Given segments A, B on the lower ray, choose a fixed unit U on the upper ray and paste their multiples mU, nU there. Then the Pythagorean dogma says: There are m, n to the effect that the segment B(nU) becomes parallel to the segment A(mU). The figure shows the segments A/m, B/n to be equal which proves C = A/m = B/n to be a common measure of A and B, since A = mC, B = nC. The square and its diagonal, or the like, were to become Pythagoras’ Waterloo. His world broke down: Given a side A and a diagonal D of the square or else of the regular pentagon, the Guru and his followers would not find them a common measure: the pairs of segments prove to be incommensurable.{1a,d,e,f} How did those people find out? If A and D were supposed to have a common measure C, then, in case of the square{1b}, a sequence of squares could be constructed such that ‒ ‒

C goes in the side and diagonal of each square, the squares become arbitrarily small,

thus leaving no room for C in the end. The event was to be called the Pythagorean catastrophe. However, the Pythagoreans had virtually “realized” the irrational number √2 (see Fig. 4.3 and Appx. D-7 for Euclid’s famous disproof). The diagonal of the unit square solves the equation x2 = 2 geometrically, whereas the algebraic solution had to wait for millenniums.

4.3 Welcome to irrationals: the complete space of real numbers

79

nU mU U A /m = B/n

A

B

Fig. 4.2. A, B commensurable.

0

1

?

ℚ+

Fig. 4.3. Irrational "?". * * * * *

Whenever it came to calculation, numbers other than 1,2, … would have to endure being called in question. Even more than the imaginary ones, the negative numbers had a hard time to get and hold their ground. In the progress of algebra and analysis, an abundance of numbers emerged from either source and found their place among the rationals, our search being called “calculating the irrationals”. Yet, concern developed for the more intricate issue: How for instance should √2 and √3 become multiplied? Dedekind laconically stated{2a}: “Nobody ever showed them to result in √6.” Are roots, in this respect, less delicate than the non-algebraic irrationals π and e? * * * {2b}

Right after 1870 , this became the challenge of the time: the rigorous foundation of analysis as concerns real numbers. They had to grow from the rationals, which was about the only consent among those who cared. Yet, in their majority those knew the way would require them to give up the traditional quality set by the rationals, their quantity in the sense of geometrical identification (cf. Figs. 4.2, 4.3). Even the rational numbers themselves must be recognized in need of a rational concept. Ancient Greeks, unless they were merchants, would deny the nonintegers the status of numbers for a long time. How seriously do we take them? It’s alright if we regard 2/4 = 1/2 as relating fractions in the sense: “two fourths make one half”. However, what is the product, what the quotient of fractions supposed to mean? Clearing such questions{3a} might help us understand the present issues and solutions. How do we appropriately interpret the equation above? To this, the fractions are considered each to form an ordered pair that represents quite a set of ordered pairs, an infinite set. The symbol ½, like (1, 2), stands for the set of all ordered pairs (m, n), m, n ∈ℕ = {1,2, …}, that share the property 2m = 1n; accordingly, the other one represents the set {(p, q) : 4p = 2q}.{3b, 4a} So, both sets are the same! Its elements (m, n), (p, q) directly compare by 2mq = nq = 2np (in elementary school we called nq a common denominator of the two fractions). This makes mq = np the relation looked for. It is said to qualify the fractions (m, n) and (p, q) to be equivalent{3c, 5a}, written (m, n) ∼ (p, q) :⇔ mq = np. A subset of ℕ × ℕ like {(m, n): 2m = 1n} is named an equivalence class on ℕ × ℕ with respect to the equivalence relation “∼”{3d}. The class is called a rational number. It can be represented by any of its fractions like is stated in the following (cf. Appx. D-8). We write and note [p, q] ≔ {(m, n) : (m, n) ∼ (p, q)}, ℚ+ ≔ {[p, q]: p, q ∈ℕ}; (m, n) ∈ [p, q] ⇔ (m, n) ∼ (p, q) ⇔ [m, n] = [p, q].

(1)

80

4 Series seriously: the Greek comet reappears

The set [p, q] is called the equivalence class defined by (p, q). Due to (1) the classes prove to be disjointed, forming a “partition”{3d} on the set of all fractions. We will see [p, p] = [1, 1], p∈ℕ, to take the role in ℚ+ that “1” plays in multiplying natural numbers. The set ℕ is to be found in ℚ+, disguised, the elements n ∈ℕ being identified with [n, 1]. One speaks of embedding ℕ into ℚ+ with more at stake than set inclusion. To what advantage does ℚ+ exceed the counterpart of ℕ? On ℕ, multiplication lacks inversion. How can multiplication on ℚ+ be organized in order to allow for division in ℚ+ and in order that multipication on ℕ becomes simulated by multiplication in ℚ+? How may the product of [p, q] and [p', q'] then be defined? Might we rule [p, q]·[p', q'] ≔ [pp', qq'] in compliance with (p, q) · (p', q') ≔ (pp', qq') as we were told by the math teacher? To this end, the result must not depend on the choices of (m, n)∈[p, q],(m', n')∈[p', q']! Indeed, due to (1), we note mq = np , m'q' = n'p' to infer mm' qq' = nn' pp' and hence (mm', nn') ∼ (pp', qq'). Multiplication on ℕ is simulated on ℚ+ by [m, 1]·[n, 1] = [mn, 1]. An equation n · x = 1 “in ℕ” has a solution only if n = 1; an equation [p, q]·[x, y] = [1, 1] always has a solution, namely [x, y] = [q, p]. With respect to the “multiplicative structures” on ℕ and on ℚ+, we say: (ℕ,·) is properly extended to, or embedded into, “the structure” (ℚ+,·). By the very same idea, (ℕ0,+) extends to (ℤ,+) (Appx. D-9). * * * * * Before, deficiencies were pointed out on two occasions: (2 i) In Fig. 4.3, Pythagoras could not fill the gap denoted "?". (2ii) Proof was missing for the sufficiency part of the Cauchy criterion in 4.1. Founding analysis means to provide the real numbers, a vital need of analysis. Satisfaction materialized in the course of the 19th century. We would phrase it thus: The ordered field ℚ of rational numbers{3e, 5b} has to be extended through a structure that, basically alike, adopts the irrationals. Fashionable didactics might readily add an axiom of completeness. Now, it is one thing to demand something, it is another to make sure some thing exists to comply with the postulate. A system of axioms must be substantiated in that they are proved to be compatible. Any concrete implementation can do the job: any model. It is giving value to what the axioms are about, like paper money gets value through gold backing. A good few models of ℝ were developed, looking rather different. To demonstrate variety, we follow, rudimentarily, the concepts{2c} of Richard Dedekind and of Georg Cantor, which pertain to (2i), (2ii), respectively. For example, we choose the line’s gap √2 in each case. (For a summary by Dedekind himself see {6a}.) * * *

4.3 Welcome to irrationals: the complete space of real numbers

81

First we will discuss Dedekind’s model.{2d, 3f, 4b, 7a} To give the basic idea, we may reduce ℚ to ℚ+ and almost disregard the operations. With ρ ∈ℚ+, both the pairs of sets Lρ ≔ {r : r ≤ ρ} , Uρ≔ { r : ρ < r} or Lρ°≔ {r : r < ρ} , Uρ°≔ { r : ρ ≤ r}

(2)

from ℚ+ follow the pattern L, U ≠ Ø, L ∪ U = ℚ+ , L ∩ U = Ø , l < u for all l ∈L, u ∈U. We call any such an ordered pair (L, U) a Dedekind cut on ℚ+, with a lower and an upper class, respectively. In particular, the cuts in (2) are called generated by the rational ρ. Contrary to the classes in (2), the sets { r : r2 ≤ 2}, { r : 2 < r2} compose a cut on ℚ+ which is not generated by some ρ ∈ℚ+, its classes being separated by a gap in the ray of positive rational numbers. As Dedekind was doing in general, we consider the set C of all cuts on ℚ+. For any two, let the relation (L, U) ≤ (L', U ') be defined by L ⊆ L'. So, cuts can be formed on C accordingly. Dedekind asserts: To every cut (L, U) on C there exists one and only one cut c ∈ C such that l ≤ c ≤ u for all l ∈L, u ∈U.{7b} Completeness, as being realized by Dedekind, takes the form of the real numbers’ “least upper bound property”{3g, 7c} (cf. (2i)). Our sketch only dealt with C as an ordered set. Operations must be conceived, compatible with one another and with the order relation. Dedekind made a virtue out of deficiency when making the gaps themselves the subjects of operation. For instance, C becomes a multiplicative group when defining the product of (L, U) and (L', U ') by L·L' ≔ { l·l ': l ∈L, l ' ∈L'}. Dedekind’s achievement invites us to identify real numbers with cuts on ℚ. He himself would not. Even this great mathematician was not open to that view: Dedekind called his non-rational cuts generated by “irrational numbers”. One may say so in the figurative sense, yet he explicitly stuck to it as show {2d, 6b, 8a}. He was criticized for this, particularly by Bertrand Russell. * * * Dedekind had considered the irrationals squeezed in between rationals. Cantor, in view of (2ii), was aiming at an extension of ℚ where every Cauchy series was procured a sum. He used the concept of equivalence classes.{4c, 5c} To motivate extension, {5d} details why c

1

c1 = 1 , cn+1 = 2n + c (n = 1, 2, …) n

(3)

forms a rational sequence which looks in vain for a limit in ℚ. It is a Cauchy sequence and therefore bounded; so, because of (3), it cannot converge to 0; it does not converge to any c ∈ℚ either, since otherwise c = c/2 + 1/c and hence c2 = 2, c ∉ℚ {1c, 3h, 5e, 7d}. (Appx. D-10) Through cn + 1/n (n = 1, 2, ... ) we also get a Cauchy sequence that does not converge in ℚ , as do all the sequences which differ from (cn) by a rational null sequence. Let C denote the collection of all Cauchy sequences in ℚ. Recall when, above, we started out from the set

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4 Series seriously: the Greek comet reappears

of all fractions (m, n) ∈ℕ×ℕ and defined their equivalence classes. Now, by analogy, equivalence on C is defined{3c, 5a} through (sn) ∼ (tn) :⇔ lim (sn – tn) = 0. Like at (1) above (cf. Appx. D-8), we define classes on C by [tn] ≔ { (sn) : (sn) ∼ (tn)} and note (sn) ∈ [tn] ⇔ (sn) ∼ (tn) ⇔ [sn] = [tn]. The equivalence classes of rational Cauchy sequences are the elements in Cantor’s model for ℝ, here likewise denoted ℝ. As to operations on this model confer the pattern for multiplication in ℚ+, detailed above: Knowing the product of Cauchy sequences to be a Cauchy sequence, we have to verify that it is correct to define the product [tn]·[tn'] ≔ [sn sn'] with whatever sn, sn' such that (sn) ∼ (tn) , (sn') ∼ (tn'). There is an absolute value on ℝ so as to define distances in Cantor’s model and form Cauchy sequences in it. To be explicit: such a sequence is denoted (Ck : k = 0,1, ... ), where Ck = [cnk] with (cnk : n = 0,1, ...) ∈C . – – The result is the completeness property of ℝ in Cantor’s diction: Theorem 4.5. To every Cauchy sequence (Ck) in ℝ there exists one and only one C ∈ℝ such that Ck → C as k → ∞. It substantiates the sufficiency part of the Cauchy criterion in the language of series and sequences. Dedekind cuts ray and line. (His concept was close to that of Eudoxus whose theory of proportions may be interpreted in terms of the order relation for our real numbers; confer {6c, 9} as to how far.) Cantor’s model is less handy. Whoever feels uncomfortable about every element of ℝ being represented by a class and every class being represented by a series, say, she or he may take comfort in that Cantor’s classes are represented by well acquainted Cauchy series: decimal fractions that is{3i}. Comfortable… However, whoever shakes his or her head about equation 3.1.2(3), he or she didn’t grasp it: the real number. The Cauchy sequences ( .9, .99, .999, ... ) and (1,1,1, ... ) being equivalent, the class they belong to can be represented either way, or just identified with 1 through embedding the algebraic structure of natural numbers into ℝ. (Apropos of he and she: In fashionable German, “Freund and Freundin” get neutralized by the ambiguous construct “FreundIn”. I won’t write “She”.) * * * * * Identification is sophistication. Mathematicians are human-minded, too. If Dedekind did not interpret his own creation appropriately, his contemporaries could hardly be expected to put up with it, let alone with Cantor’s model. Emotions arouse. Those marvelous π and e are

4.3 Welcome to irrationals: the complete space of real numbers

83

said to be classes of series? Analysis once grew out of geometry, since then it enjoyed support from geometry. Does it renounce that heritage? Or ignore Newton’s kinematics? Of course it does not. Yet, emancipation from geometry and science were necessary when it came to foundation, and this could but have been performed through modeling. Now, that it had happened, analysis was thought purged once and for ever, having achieved a firm stand of its own. It does not matter whether some disliked the new look. However, substantial criticism would arise and matter. In a canonical fashion, the rational numbers could be “reduced” down to the integers, Giuseppe Peano gave the axioms for natural numbers which the logician Gottlob Frege created out of the empty set… So one might think to have done the very best.{8b} One’s very best, at least and maybe at most, as was given a hint at 4.1. Whatever model was provided to found the real numbers upon, it was upon infinite sets. What in the end could real numbers really count on? How could one be sure of existence at all where the infinite is involved? When David Hilbert gave his famous Paris speech (see 1.2) on what the century ahead was waiting for to be done, he coined what became his epitaph: “Wir müssen wissen, wir werden wissen.” [We must know, and we will know.]{10a} (On the Hilbert Program see {12}.) Soon after, it became clear that mathematicians were facing problems that their mathematics would not solve, questions denying Yes and No{12a}. Euclid’s axioms had been regarded as self-evident; the nature of axioms changing, they survived by the axiomatic method. Yet, the infinite kept lurking. Hilbert’s dream ended when Kurt Gödel made certain that there is no overall certainty. His was the proof that, for any “appropriate” kind of set theory, completeness{10b, 12b} of axioms is not possible (named “theorem of incompleteness”). After analysis had suffered the loss of geometrical evidence it now was being endangered by fundamental skepticism. The platonist, while convinced that mathematical objects have an existence of their own, finds himself opposed to so-called formalism{12c} and constructivism{12d} which account for a loss that is to be addressed under headings like “From Certainty to Fallibility”{12e},“Foundations, Found and Lost”{12f}. (See the cover of {13}, top side, but don’t believe in every word.) In the meantime, competing objections have become part of analysis’ history. Relativity had entered mathematics: in analysis. Yet, apart from fundamental rejection and individual ambiguity, its mainstream would follow what had been achieved by 1900. Real numbers remained grounded on their axioms, like classical electrodynamics is grounded on the Maxwell equations however these might once have been grounded. {1} {2} {3} {4} {5} {6} {7}

RADEMACHER; TOEPLITZ: {1a} 23–25. {1b} 24, Fig.13. {1c} 25 bottom. {1d} WUẞING [1] 57 f. {1e} SONAR 24–26. {1f} KÖRLE 8 f. EDWARDS JR.: {2a} 329. {2b} 330 bottom. {2c} 331–333. {2d} 331. KRANTZ: {3a} 43–50. {3b} 43. {3c} 37. {3d} 38. {3e} 46, 49. {3f} 63. {3g} 51, 64 f. {3h} 50 f. {3i} 1. HOBSON [1]: {4a} 15–20. {4b} 23–27. {4c} 28 ff. ENDL; LUH: {5a} 40–42. {5b} 10 f, 15 f. {5c} 43 ff. {5d} 38 f. {5e} 39. BECKER: {6a} 224–245. {6b} 244. {6c} 241 ff. RUDIN [1]: {7a} 3–10. {7b} 9 f. {7c} 10 f. {7d} 2.

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4 Series seriously: the Greek comet reappears

{8} {9} {10} {11} {12}

HEUSER [2]: {8a} 700 middle. {8b} bottom. BOURBAKI [2] 144, 145 f. KÖRLE 86, 88. GÖDEL: {10a} 208 middle. {10b} 203. SONAR 577 ff. DAVIS; HERSH: {12a} [1] 320; [2] 336. {12b} [1] 228; [2]236. {12c} [1] 319 f; [2] 335 f. {12d} [1] 333 ff; [2] 351 f. {12e} [1] 317–359; [2] 333–379. {12f} [1] 330– 338; [2] 347–356. {13} KLINE [2].

4.4

Another complete space: the home of convergent sequences

When Cauchy wanted to say that a real series converges absolutely, he wrote: It remains convergent after all negative terms have changed to their numerical values.{1} We use a simple symbol to make Cauchy’s numerical value of a difference indicate the distance between real numbers, even including complex ones. (Leibniz shows us that symbols may be more than a matter of convenience.) Cauchy spoke of “distances becoming arbitrarily small”. He would not address them by name: not by value, he did not relate distances. Weierstraß did. The mapping x ↦ √ x2 = | x |, defined on the number field ℝ, measures the distance | x – 0 | between the vectors x and zero of the linear space ℝ1. It is called a norm on ℝ1 and makes ℝ a complete normed space (4.3), a “number space”. This readily transfers to ℝ2, ℝ3, ... . (For notions and notations, particularly for the norm postulates, see {2a, 3a, 4, 5a, 6a}.) The Euclidean norm (x, y) ↦ √(x2 + y2) on ℝ2 and its extension to any ℝn became first choice, furnishing the Euclidean distance each (Appx. D-11). In function spaces integration correspondingly provides π

the Euclidean norm √ ∫ –π f (x)2 dx {2a}, generated by a “scalar product” (cf. 3.3.4(3..)). Now, (x, y) ↦ || (x, y) || ≔ max (| x |, | y |) creates a norm and thus a distance on ℝ2, too. On Euclidean and maximum norm coincide, on the higher number spaces both kinds of norms prove equivalent in the sense of {3b, 5b}. The maximum norm on any ℝn has a counterpart on the linear space of real functions that are continuous on [0, 1], namely f ↦ max {| f (x) |: x∈ [0, 1]}{2a.5c}. Being entitled supremum norms (“sup norm”), the maximum norms on the spaces ℝ1,

ℝn naturally extend to any linear subspace B ⊆ℝ∞ of bounded real sequences, with the norm of x = (x0, x1, … ) defined by || x || ≔ sup { | xk |: k = 0,1, … } ≡ supk | xk | , x∈B.

(1)

(That looks no other in “complex number spaces” (cf.{4}) where many results we are going to discuss or prove hold as well. Where we label a theorem “for reals” that might only point to the context in our exposition.) As was the case with the real number space ℝ (4.3), Cauchy sequences in any normed space are defined according to 4.1 and ask for a limit within that space. To illustrate convergence of Cauchy sequences in some B, we may refer to Fig. 4.1, case u < 1, for analogy. The

4.4 Another complete space: the home of convergent sequences

85

place of the functions fn : x ↦ fn(x) ∈ℝ, 0 ≤ x < u, where n ∈ℕ0, is now being taken by functions φn: k ↦ φn(k) ∈ℝ (k∈ℕ0), where n∈ℕ0 (“sequences” that is, cf. 3.1.1). Remark. We make Fig. 4.1 correspond to the matrix (φn(k): n,k = 0,1, …) by running its row index n. In the sequel, we will also run it by its column index k, thus forming the sequences ψk ≔ (φn(k): n = 0,1, …), k = 0,1, … .

(2)

We first give the analogue to Def. 4.3. The sequences φn = (φn(k): k = 0,1, …), n = 0,1, … , converge to the sequence φ uniformly as for k = 0,1, … if, for every ε > 0, there is some nε such that |φn(k) – φ(k) | ≤ ε holds for every n ≥ nε and any k ∈ℕ0. In terms of (1) that is supk |φn(k) – φ(k) | ≕ ||φn – φ || → 0 as n → ∞ . Theorem 4.6. Any linear space B of real bounded sequences is complete under the sup norm (1). Proof. Let φn form a Cauchy sequence in B, which means: given ε > 0, there is n(ε ) such that || φn – φn+p || ≤ ε for n ≥ n(ε ), p∈ℕ0, in detail given ε > 0, there is n(ε ) such that | φn(k) – φn+p(k) | ≤ ε for n ≥ n(ε ), p∈ℕ0 and all k = 0,1, … .

(3)

The hypothesis (3) includes, with any fixed k ∈ℕ0, that (2) defines a Cauchy sequence ψk in ℝ; this provides numbers limp → ∞ φn+p(k) ≕ φ(k) ∈ℝ, k∈ℕ0. It results in given ε > 0, there is n(ε ) such that |φn(k) – φ(k) | ≤ ε with n ≥ n(ε ) and all k = 0,1, … (4) For one thing, (4) shows φ = (φ(k) : k∈ℕ0) ∈B since |φ(k) | ≤ |φ(k) – φn(ε )(k) | + |φn(ε )(k) | ≤ ε + || φn(ε ) ||, k∈ℕ0 . Finally, (4) yields supk |φn(k) – φ(k) | = ||φn – φ || ≤ ε for n ≥ n(ε), that is φn → φ ∈B as n → ∞. – – * * * When we consider the normed space C of all convergent real sequences, Thm. 4.6 asserts that every Cauchy sequence in C has a bounded sequence for limit. Is it convergent? Quite similar as in the proof of Thm. 4.4 we can argue to arrive at Theorem 4.7. The linear space C of real convergent sequences is complete under the sup norm (1).{4}

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4 Series seriously: the Greek comet reappears

Proof. Let φn ∈ C form a Cauchy sequence. By Thm. 4.6 we know that || φn – φ || → 0 holds with some bounded sequence φ. We have to show φ ∈ C which is to say that φ is a Cauchy sequence in ℝ. Given ε > 0, there is n(ε ) such that, with some φn(ε), we get |φ(k) – φ(k+p) | ≤ |φ(k) – φn(ε)(k) | + |φn(ε)(k) – φn(ε)(k+p) | + |φ n(ε)(k+p) – φ(k+p) | ε

≤ ||φ – φn(ε) || + |φn(ε)(k) – φn(ε)(k+p) | + ||φn(ε) – φ || ≤ 3 3 for k sufficiently large and all p ∈ℕ0. – – The great mathematician Stefan Banach{2b} was the first to publicly introduce the concepts of functional analysis{2c} (cf. 3.3.4). He did so in his doctoral thesis of 1920{6a} followed by a most incentive treatise{6b}. The so-called Polish School he initiated remained active for nearly two decades. To the group the Nazi terror was fatal; Banach narrowly escaped murder but also became a victim{2d}. In his honor, complete normed spaces are called Banach spaces.{2e, 3b, 4, 5d} * * * We are now prepared to describe limit and limitation within a general context. At present (further cf. 5.3.2), we consider the linear space of convergent real sequences, denoted C. The vectors x = (xn) ∈C and the scalars α ∈ℝ join in the exterior operation α s = (α xn) ∈C; the scalar field accounts for C being called a real vector space, or a vector space over ℝ. Supremum norm and the limit it provides “operate” on C, assigning scalars to vectors; any mapping φ: C → ℝ is called a functional{2f} on C. Norms appropriately furnish the spaces of sequences as show the continuous functionals. Theorem 4.8. (N ) The supremum norm is a continuous functional on any linear space B of bounded real sequences. (L ) The limit is a continuous functional on the linear space C of convergent real sequences. Proof. (Cf. Appx. D-12.) Let x ↦ || x || , x∈B, as defined in (1). (N ) Let N : x ↦ N x = || x ||, x∈B. If xk → x then N xk → N x as k → ∞ because | Nxk – N x | = | || xk || – || x || | ≤ || xk – x ||. – (L ) Let L: x = (x0 , x1 , … ) ↦ L x = lim xn, x∈C. If xk → x then L xk → L x as k → ∞ because | L xn – L x | = | L ( xk – x) | = | limn (xk – x)n | = limn | (xk – x)n | ≤ supn | ( xk – x)n | = || xk – x ||. – – Contrary to L on C, the functional N on the space of all bounded sequences is not linear. (Why?) * * *

4.4 Another complete space: the home of convergent sequences

87

Any vector space ℝn has a base, for instance the “canonical” one: e1 = (1,0, ... ,0), … , en = (0,0, ... ,1); each element x = (x1 , x2 , ... , xn ) ∈ℝn is being uniquely represented by x = x1 e1 + x2 e2 + ... + xn en. Is there an analogue for the Banach space C of all convergent sequences? Yes. For simplicity we will confine ourselves to the subspace C0 of null sequences. With every x = (x1 , x2 , ...) ∈C0 there holds x = x1 e1 + x2 e2 + … , where ek = (δik : i = 1,2, … ), k = 1,2, … ,

(5)

which is defined and proved by || x – (x1 e1 + … + xk ek ) || = || (0,0, … ,0, xk+1, xk+2, … ) || = supl > k | xl | → 0 as k → ∞, (6) since x ∈C0. The approach in (6) is called sectional convergence.{3c, 7} (As for uniqueness of (5) to represent x = (x1, x2, ...), do Appx. D-13.) {1} {2} {3} {4} {5} {6} {7}

CAUCHY 128 f. SONAR: {2a} 594. {2b} 598–602. {2c} 592–595. {2d} 601 f. {2e} 595. {2f} 595 bottom. ZELLER; BEEKMANN: {3a} 20 f. {3b} 21. {3c} 42 ff. POWELL; SHAW 163. LANG: {5a} 18–20. {5b} 20. {5c} 19. {5d} 65 ff. {6a} SIEGMUND-SCHULTZE 502. {6b} BANACH. BOOS 355.

5

On the verge of summability

5.1

Divergent series: suspected and respected

Newton gained a lot when he treated series of powers like polynomials. Euler gained no less when he treated infinite series algebraically, a method immune to analytical criticism. In view of his accomplishments, shouldn’t he be awarded the license to have done so? Regardless of convergence? James Gregory gave convergence its name, but didn’t care about a general concept. Leibniz, likewise conscious of the process, took a liberal attitude. Euler was thoughtful enough to formulate the Cauchy postulate (see 4.1). To him convergence was just one aspect of infinite series and not the major one. Those days, it was rather perceived a matter of intuition that was nourished by paradigms like the geometric series or the Leibniz criterion for alternating series. Euler’s manipulations must have been judged a relapse behind the epoch of Newton and Leibniz, not even a comparison with the work of Eudoxus and Archimedes. When, by the end of Euler’s century, the subjects of analysis became increasingly questioned, divergent series were the first to be put on the index. In a letter, 1826, Abel{1} bitterly complains about series: He points out that, excepting the geometric one, there were hardly any whose sum had been determined rigorously. His words, translated: “What is most important in mathematics lacks justification.” In particular: “On the whole, divergent series are devilries [‛Teufelszeug’], and it is a shame that people venture to ground a proof on them. By their use one can get whatever is wanted.” However, Abel must concede Euler’s great contribution and he wondered how such poor reasoning could give that rich a harvest; it would be most worthwhile to find out why. Abel died at the age of 27. He could have accomplished that much, including an analysis of “Teufelszeug”. Cauchy had condemned divergent series, too, yet without zeal. Did he guess something cryptic about them of the sort that Euler{2} might have expected to be revealed sometime? In view of the many surprising results divergent series accounted for, one could not do away with them in total. * * * Now, that convergence had been consolidated, why shouldn’t divergence be given a chance on a rigorous treatment? Euler had assigned values to divergent series ad hoc when they came across (3.2.6), not without method. His must not be taken for what later will be called a method of summability. Still, in this very sense, an “Euler method” was posthumously conceived, i.e. a series transformation that once was intended to accelerate convergence. (We noted that Nīlakantha had also been engaged in this; cf. 3.2.1.) Euler seemingly would not apply it to series that do not converge at all.

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5 On the verge of summability

It is a different thing altogether that made its appearance when, by the end of the 19th century, Ernesto Cesàro observed what strongly suggested to consider series more comprehensively: a most creative perspective by the name of limitation. In a multitude of ways and on as many reasonable grounds series were assigned “sums” other than by accumulation. The title of the theory, summability, is due to history and might better spell limitability, if not a tongue-twister in any language. The basic idea of limitation is mapping. Series, sequences and whatever get transformed in order to ask for the properties of the transform, for convergence in particular. Preferably, those mappings ought not spoil a convergent series’ image: a regular method has to preserve convergence and limit. For an example and for the sake of didactics, Otto Szász{3} turned a sequence s0, s1, ... into the sequence σn = (sn−1 + sn)/2, n∈ℕ0 (s–1 ≔ 0). This method clearly is regular and proves efficient in that it applies to the divergent sequence sn = 1,0,1,0, ... which, by σn ≡ ½, qualifies for the generalized limit "lim" sn ≔ lim σn = ½. (As to “efficiency”, confer 5.3.1.) Our survey will be exemplary throughout, confining itself, except for one contrasting example, to the first and foremost two methods in history. They have become paradigms, one of which being called{4} the most natural one. Convention. As is common practice, numbers will mean complex ones here unless specified otherwise (like by order relation, e.g.). {1} {2} {3} {4}

LÜTZEN 222 f. SPEISER VII middle. SZÁSZ 4. BOOS 8, 14. COOKE; ARNETT 212. ZELLER; BEEKMANN 111.

5.2

The initiation of Cesàro and Abel summation

5.2.1

Grandi’s series, recurrent series

In 3.2.6, we mentioned Guido Grandi and his series{1a, whether the series

2a, 3a, 4a}

. That learned monk asked

1−1+1−+…

(1)

through 0 = (1 − 1) + (1 − 1) + … = 1 − (1 − 1) − (1 − 1) − … = 1

might tell us how God had created the world out of nothing.{1b} Math students are trained not to believe in infinite series considered as sums. Grandi apparently did, at least he thus treated them. If he had valued the series by way of s ≔ 1 − 1 + 1 − + … = 1 − [1 − 1 + 1 − + …] = 1 – s {5a, b}

(2)

he readily would have arrived at 1 1−1+1−+…= 2.

(3)

5.2 The initiation of Cesàro and Abel summation

91

Instead, Grandi specified (3) from 1 + x + x2 + … =

Σ∞ xk = k=0

1 1– x

(4)

where he either thought of x ↓−1 or – as is suspected in {1c} – just plucked in x = −1. And he provided more speculation. Grandi wrote a book; the Marburg professor Christian Wolff shook his head and sent a letter to Leibniz. It was the first “paper” on summability. Leibniz’ answer was the second, published in 1713{1b}. Jacob Bernoulli{1d} and Leibniz{1e, 4b} seriously discussed Grandi’s valuation (3) by further arguments. Those days, mathematicians would not ask how to define (1) but rather “what is (1)?” * * * If Grandi’s oscillating series (1) should be valued at all, ½ surely figured as the most reasonable sum{5a}, at least the most natural{5b}. After having studied series like (1) systematically, Jacob Bernoulli called (3) a result to go by “paradox non inelegant”{1f}. There hardly was anything which Leibniz was not interested in. It’s not like Leibniz to have overlooked trick (2) above; presumably he would rather explain than presuppose a sum to exist. He furnished two arguments.{1g} The first pertains to what is nearest at hand, (4) that is. Leibniz the philosopher had maxims. His lex continuitatis (law of continuity) reads: What is true up to the limit is true at the limit.{1g.5c} (Students would object by 1/n > 0 = lim 1/n.) Leibniz felt uneasy about the detour via a power series, the “sum value” of the series should result from its data itself. The partial sums of Grandi’s series, he pleaded, are assuming the values 1 and 0 with “equal frequency”; hence, the sum is fairly their mean on grounds of probability.{1h, 5d, 6a} Euler became involved, too (cf. 3.2.6). Since the series owns a general term, only the what and the why of the value are what matters. His philosophy on this was Def. 3.2 which left no doubts. In fact we can read what Euler explicitly says about Grandi’s series{7a}, translated: “If, in the course of calculation, I am coming on the series and substitute for it no value other than ½, surely nobody will blame me for an error.” There was only one who was strictly opposed to any valuation of the Grandi series: Pierre Varignon. He pointed out that the series’ terms need to decrease, so much so in order to render “the reminder negligible”.{1i} * * * Later on, Grandi’s series will be called periodic or recurrent{7b} in the sense of ∞

1 − 1 + 1 − + … = Σk=0 ak , where ak = ak+2 , a2n + a2n+1 = 0 (k, n = 0,1, …)

with a0 + a1 its period. The valuation (4) is matched, according to Leibniz, by the means of n subsequent partial sums sn = Σk=0 (–1)k, namely half of s2n + s2n+1 = s2n+1 + s2(n+1) ≡ 1: ∞

1

1

lim x↑1 Σk=0 (− x)k = 2 (s0 + s1) = 2 (1 + 0) , n = 0,1, … .

(5)

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5 On the verge of summability

Equation (5) was suggested extending to recurrent series in general, by Johann Bernoulli’s son Daniel{3b} in 1771: n

Theorem 5.1. Let sn ≔ Σk=0 ak. With an integer p ≥ 2, let ak be such that ak = ak+p for all k = 0,1, … and

∞ k Σp–1 k=0 ak = 0. Then, the series Σk=0 ak x converge for 0 < x < 1, resulting in ∞

lim x↑1Σk=0 ak xk =

1 p

Σp–1 n=0 sn.

(6)

It was confirmed by Lagrange (1799){3c} and by Raabe (1836){3c, 6b}. ∞

As being reported in {3b}, a series Σk=0 ak which meets the assumptions of Thm. 5.1

was denoted recurrent even though, long ago, de Moivre{2c, 7c} had used the word in a different sense. Recurrent series do not diverge breathtakingly. Nevertheless they are a first example for divergence to become the subject of a proper proof. REIFF: {1a} 65 ff. {1b} 66. {1c} 65. {1d} 56 f. {1e} 66–68. {1f} 57. {1g} 66 f. {1h} 67 f. {1i} 69 f. CAJORI [1]: {2a} 238. {2b} 230 above. {2c} 230 middle MOORE: {3a} 1. {3b} 3. {3c} 4. JAHNKE [1]: {4a} 154 f. {4b} 155. HARDY: {5a} 2. {5b} 6. {5c} 14. {5d} 13 f. RAABE: {6a} 355. {6b} 356 f. EULER: {7a} [4] 593 top (Latin). {7b} [1] 229 ff; [3] 181 ff. {7c} [1] 78 ff; [3] 53 ff (54 line 4, read “is” for “if ”).

{1} {2} {3} {4} {5} {6} {7}

5.2.2

Frobenius’ theorem and the limit theorems of Cauchy and Abel

Grandi’s paradox was to become a productive incentive. Next to Thm. 5.1, in 1880, Georg Frobenius wrote a paper “On the Leibnitz [sic] Series” {1a}. The author calls his theorem a conjecture by Leibniz. In fact, the title refers to Grand’s series rather than to what we call Leibniz’ series, i.e. 3.2.1(1), and is due to how Leibniz engaged for the valuation of that series. Leibniz’ main concern in this was the recourse to “probability”. In his (Latin) letter mentioned in 5.2.1, he concedes that, to bridge his argument to 5.2.1(4), he makes use of metaphysics rather than mathematics.{1a} Nevertheless, Frobenius gives credit to Leibniz. We cannot help but interpret Frobenius as follows. The oscillation of Grandi’s partial sums ν

sν = Σk=0(−1)k =

1 (1 2

+ (−1)ν )

suggests to be brought under control by means of the arithmetic means 1 1 1 1 1 n n n ν n + 1 Σν=0 sν = 2 n + 1Σν=01 + 2(n + 1) Σν=0 (−1) ≕ 2 + Rn, where Rn → 0,

(1)

5.2 The initiation of Cesàro and Abel summation

93

hence by another kind of arithmetic average than on the right of 5.2.1(6), explicitly mentioned by Frobenius{1a}. He presents the theorem of his own paper saying “Hence Leibniz maintains [‘Leibnitz behauptet also’], though without proof, the following, a generalization of a well-known theorem of Abel’s.” {2} Assuming “all numbers to be real” he proves: n

Theorem 5.2. Let sn≔ Σk=0 ak (n = 0,1, …). Suppose that 1

n

(i) : the sequence n + 1 Σν=0 sν converges, ∞

then

(ii1) : the series Σ ak xk, 0 < x < 1, converge and lim x↑1Σk=0 ak xk exists

and

(ii2) : lim x↑1 Σk=0 ak xk = lim n→∞ n+1 Σν=0 sν.

1

∞

n

For a comment on Frobenius’ proof see Appx. E-1. If we suppose the sequence sn itself to converge then Abel’s limit theorem Thm. 4.2 covers (ii1) in Thm. 5.2. It also covers (ii2) by virtue of Cauchy’s Limit Theorem, i.e. 1

n

Theorem 5.3. Let σn≔ n+1 Σν=0 sν. (i) If sn → λ then σn → λ ;{3} (ii) convergence of σn does not infer convergence of sn. ε

Proof. (i). Take λ = 0 (see Appx. E-2). Given ε > 0, choose nε such that | sν | ≤ 2 for all n

n

ν ≥ nε, write Σν=0 sν ≕ M(ε) + Σν = n sν. With n ≥ nε sufficiently large, we thus get | σn | ≤ ε 1 ε n + 1 | M(ε) | + 2 ≤ ε . –

(ii). See σn in the case of Grandi’s series, at (1). – – n

∞

Abel’s limit theorem asserts that Σk=0 ak → λ infers lim x↑1Σk=0 ak xk = λ. This, by Cauchy’s limit theorem, is a result strictly weaker than Frobenius’ Thm. 5.2. By the latter, we will see the leading protagonists of early summability introduced. The paper became the herald on top of a list{4} which by its length gives a lower estimate for the abundance of results in less than a centennial. Otto Hölder refers to that theorem, being the first who, in 1882, systematically{5} employed arithmetic means as a means of sequence transformation. {1} {2} {3} {4} {5}

FROBENIUS: {1a} 262. {1b} 263 line 6. MOORE 4 middle. CAUCHY [1] 63 (Théorème III). KNOPP [1] 73 f; [2] 72 f. ZELLER; BEEKMANN 194–301. HÖLDER 536.

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5 On the verge of summability

5.2.3

The Cauchy product revisited by Abel and Cesàro

In a way, Abel also proved a promoter of Cesàro’s method. The story begins with Abel engaging in the Cauchy product of series. At 3.4.2, Cauchy’s assumptions in Thm. 3.4 resulted in absolute convergence of the product series. What may be sufficient for ordinary convergence? The proof of Mertens’ theorem Thm. 3.5 requires but one factor to converge absolutely (Appx. C-16). Might one dispense altogether with assuming absolute convergence in order to render, so to speak, the Cauchy product converge to the product value of its factors? The example 3.4.3(9) shows that even an absolutely convergent Cauchy product may have divergent factors. Abel provided a remarkable contribution when he used his Limit Theorem (Thm. 4.2) for an elegant argument. Well, he couldn’t do without any absolute convergence; we will see him raise a loan with power series. Theorem 5.4 (Abel). All three series being assumed convergent, their values result in l ∞ ∞ (Σ∞ a )(Σk=0 bk ) = Σl=0 Σi=0 ai bl – i.{1, 2a, 3a, 4a} i=0 i

(1)

(My doctoral supervisor put it thus: “Cauchy products rather diverge than converge to the wrong value.”{4a}) Proof. With reference to 3.2.4(4), like in the proof of Thm. 4.2, we note absolute convergence of ∞ Σ∞ a xi, Σk=0 bk xk, i=0 i

0 < x < 1,

which, by Cauchy’s Thm. 3.4, results in ∞

∞

(Σi=0 ai xi )(Σk=0 bk xk) =

l Σ∞ (Σ a b ) xl, l=0 i=0 i l – i

0 < x < 1.

(2)

Abel’s theorem when applied through lim x↑1 to all three series in (2) yields (1). – – * * * In his inspiring paper, Cesàro expressively referred{5a} to Abel’s proof above which he apparently considered detouring the aim, all the more as absolute convergence was instrumental. Cesàro might, or is rather likely to have studied the role which Cauchy’s limit theorem plays within Frobenius’ result Thm. 5.2, for it improves upon Abel’s limit theorem. Here is, how Cesàro obtains Thm. 5.4 “directement”{5a}, namely improving Thm. 5.4 by way of Thm. 5.5 below. Cesàro notes{5b}, and {4b} reads the same, that the partial sums An, Bn, Cn of the series in (1) are related by

Σnν=0Cν = Σnν=0 Aν Bn–ν.{2b} (For the technicalities, see {5c}, Appx. E-3.) Suppose that An → A, Bn → B, note that 1 n 1 n 1 n n Σν=0Cν = n Σν=0(Aν – A) Bn–ν + A [n Σν=0 Bν]= In + IIn

5.2 The initiation of Cesàro and Abel summation

95

with IIn a null-sequence by Cauchy’s limit theorem Thm. 5.3. For the proof of In → 0, let | Bn | < β (n = 0,1, ... ). Given ε > 0, there is n(ε) such that | Aν – A | ≤ ε /(2β) for all ν > n(ε). Now, for all sufficiently large n > n(ε) we get B

| In | ≤ | Σ0 ≤ ν ≤ n(ε)(Aν – A) nn–ν | +

n – n(ε) ε ε ε n 2β β ≤ 2 + 2 ,

since (Aν – A)Bn–ν /n (n = 0,1, ... ) is a null-sequence for any ν = 0, ... , n(ε), and so is their sum. Therefore 1

lim n Σν=0Cν = lim IIn = AB {5d} n

and thus Thm. 5.4, again by Cauchy’s limit theorem since (Cn) was supposed to be convergent. – – In particular, and this being the point right now, Cesàro generalized Abel’s result Thm. 5.4 on the Cauchy product as follows: ∞

∞

Theorem 5.5 (Cesàro). Let the series Σi=0 ai, Σk=0 bk converge; let C0, C1, ... denote the 1

n

∞

∞

partial sums of their Cauchy product. Then n+1 Σν=0Cν → (Σi=0 ai ) (Σk=0 bk ) as n → ∞ . Within that same paper {5}, Cesàro furnishes the most comprehensive extension of Thm. 5.5 after having generalized arithmetic means (5.3.4, 5.3.5). {1} {2} {3}

ABEL [1] 316–318 (317 bottom, f). HARDY: {2a} 228, Thm.162. {2b} 229 (10.3.1) with r = s = 0. KNOPP: {3a} [1] 331; [2] no.189. {3b} [1] 330 f; [2] no.188. {3c} [1] 331 bottom; [2] no.189. {3d} [1] 331f ; [2] no.189. PEYERIMHOFF: {4a} 3. {4b} 2. CESÀRO: {5a} 115 top. {5b} 116, following (4). {5c} 116 (4). {5d} 116 (5); read what follows as “W does not differ from the left of (5)”.

{4} {5}

5.2.4

Introducing the methods of basic Cesàro and of Abel means

The limit theorems of Cauchy and Abel, together with Frobenius’ theorem, may be regarded the overture to summability. These theorems relate the two basic methods to convergence and to one another. Throughout here, we meet them both, infinite series and sequences, so it is worthwhile to have translations at hand. Note the following (which also holds “in ℂ”) and their verification. 1

n

n

k

(1) n+1 Σν=0 sν = Σk=0(1 − n+1) ak ; (2)

∞ ∞ Σk=0 ak xk = (1 − x)Σn=0 sn xn ,

| x | < 1, whenever one of the series converges.

96

5 On the verge of summability

As to equation (1), we may start from the left, applying Abel’s partial summation (cf. 4.2.3(2)) n

to Σν=0 1·sν. Otherwise, when starting from the right, we proceed by k

n

a0 + Σk=1(1 − n+1) (sk – sk–1) =

k k+1 n n Σk=0 (1 − n+1) sk – Σk=0 (1 − n+1) sk

1

n

= n+1 Σν=0 sν.

At (2), when assuming convergence on the right, we readily get ∞ ∞ Σn=0 sn xn − Σn=0 sn xn+1 =

∞

∞

s0 + Σn=1(sn – sn–1) xn = Σk=0 ak xk ;

assuming convergence on the left of (2), we might employ Mertens’ theorem (Thm. 3.5; or consult 3.2.4(4) ) when forming the Cauchy product 1 ∞ k 1 – x Σk=0 ak x =

n ∞ ∞ ∞ ( Σk=0 ak xk ) ( Σl=0 1·xl) = Σn=0 ( Σk=0 ak ·1n–k) xn ∞

= Σn=0 sn xn , | x | < 1. * * * Let us summarize what leads up to the definitions in question. Cauchy’s limit theorem Thm. 5.3, a plain yet incentive observation, would help improve upon two results of Abel. The first one is Abel’s limit theorem Thm. 4.2 when being generalized by Frobenius’ theorem Thm. 5.2. The other one is Abel’s Thm. 5.4 on the Cauchy product 5.2.3(1); the alternative{1a} which Cesàro furnished to Abel’s argument improves upon Thm. 5.4 and further generalizes it largely in the course of Cesàro’s theory (5.3.5). Cesàro’s trick was to be called Cesàro’s method though being shared with Hölder. They both extended Cauchy’s transformation (sn) ↦ (σn) in Thm. 5.3{2} so that it became both their method of order 1. We let “C1” stand for the Cesàro method of arithmetic means{1a, 3a, 4a, 5a, 6a}; it is defined in the following ways. 1

n

n

Definition 5.1. Let sn ≔ Σk=0 ak , σn ≔ n+1 Σν=0 sν (n = 0,1, …). (i) The sequence sn is called limitable (by) C1 to λ if the sequence σn converges to λ. ∞

(ii) The series Σk=0 ak is called summable (by) C1 to ς if (ii1) the sequence sn is limitable (by) C1 to ς ; n

k

(ii2) the sequence σn = Σk=0 (1 − n+1) ak converges to ς. (Cf. (1).) With (sn), (ak) in the role of one-column matrices, the transform σn is represented in terms of the matrix products 1

(σn) = C1◦(sn) = C1*◦(ak), where C1 = (cnν), cnν = n+1 (0 ≤ ν ≤ n), cnν = 0 (n < ν), and k

C1* = (cnk*), cnk* = 1 − n+1 (0 ≤ k ≤ n), cnk = 0 (n < k).

5.2 The initiation of Cesàro and Abel summation

97

These triangular matrices represent linear mappings operating on the linear space of all sequences in ℂ as shows, for instance, C1◦(sn + sn') = C1◦(sn) + C1◦(sn'), C1◦(α sn) = α [C1◦(sn)]. (C1 is therefore called a matrix method{6b}.) * * * Abel’s limit theorem (Thm. 4.2) gives rise to “the method A of Abel means”{1b, 3a, 4b, 5d, 6c}: ∞

Definition 5.2. Let α(x) ≔ Σk=0 ak xk formally. ∞

(i) The series Σk=0 ak is called summable (by) A to ς if α(x), 0 < x < 1, converge and lim x↑1 α(x) = ς.{3b}

(ii) The sequence sn is called limitable (by) A to λ if ∞

(ii1) Σn=0(sn – sn–1) (where s–1 = 0) is summable A to λ. ∞

∞

(ii2) Σn=0 sn xn , 0 < x < 1, converge and lim x↑1 (1 – x)Σn=0 xn sn = λ. (Cf. (2).) Example. When applying Def. 5.2(i) to 5.2.1(4), the terms of Grandi’s series become the ∞

weights of its A-transform Σk=0 (–1)k xk, according to {7}. (Do Appx. E-4.) Def. 5.2 is based on (i) a series-to-function and (ii) a sequence-to-function mapping. Again, the transforms allow for matrix representations. To realize “x↑1”, all sequences xi↑1 are considered to form ( α(xi) ) = A◦(ak) = A*◦(sn) , where A = (xi k : i,k = 0,1, ... ) and A* = ( (1 − xi) xi n : i,n = 0,1, ... ) . ∞

∞

(Note that, by assumption, the “matrix products” Σk=0 xi k ak and Σn=0 (1 − xi) xi n sn, i∈ℕ0, are defined.) With respect to the discrete limit “xi↑1”, one speaks of a semi-continuous matrix transform.{5c} Remarks. n 1. In Def. 5.1, the “weighted” sum Σν=01·sν of s0, s1, … , sn is put into ratio with the sum

Σνn=01 = n + 1 of the weights. In Def. 5.2(ii2), accordingly, the weighted series Σ∞ xn sn of n=0

the sequence sn is put into ratio with the infinite sum of weights xn.{7}

2. To motivate C1 and A, reference is given on {8} to “Frobenius”. What can be read there, in two places, can hardly be interpreted to represent Frobenius’ theorem. 3. As has been done in Defs. 5.1 and 5.2, a distinction between limitable and summable would not be followed thoroughly, if not considered pedantic. Theory of summability means the same as German Limitierungstheorie. * * *

98

5 On the verge of summability

The methods C1, A when applied to convergent objects are preserving convergence and limit value, by their origin from the two Limit Theorems ahead. They are called regular or permanent.{ 3b, 5e,f} Moreover, in case of real terms, we note Theorem 5.6. Let sn, ak ≥ 0 and sn → ∞, Σak = ∞. Then (i)

1 n

Σνn=0 sν → ∞.

∞

(ii) either there is ξ ∈(0, 1) such that Σk=0 ak xk → ∞ for ξ < x↑1 or, for some ξ ∈(0, 1), there holds

Σ ak xk = ∞ for all x ∈[ξ, 1).{3b} (For proofs see Appx. E-5.) The theorem makes C1, A be called totally regular methods. For this reason, the harmonic series is not summable C1, A. To be cured by these methods, divergence needs oscillation, to a certain extent. {1} {2} {3} {4} {5} {6} {7} {8}

PEYERIMHOFF: {1a} 2 f. {1b} 24. CESÀRO 118 f. HÖLDER 536. HARDY: {3a} 7. {3b} 10. KNOPP: {4a} [1] 481; [2] no.265,1. {4b} [1] 484; [2] no.265, 4. ZELLER; BEEKMANN: {5a} 100. {5b} 6. {5c} 8. {5d} 110 bottom. {5e} 5. {5f} 59 bottom. BOOS: {6a} 8. {6b} 14. {6c} 10, 15. KOREVAAR [2] 1. FORD iv, 76.

5.3

Features of Cesàro and Abel means

5.3.1

Inclusion, limitation, efficiency

Thm. 5.2(i),(ii1) previously related the first two methods of summability: If “(sn) is limitable C1”, then “(sn) is summable A”. For short, the conclusion is given the form “C1 includes A”. With D(C1), D(A) denoting the sets of (say real) sequences summable C1, A, respectively, this logical inclusion turns into the set inclusion D(C1) ⊆ D(A), spelled “D(A) includes D(C1)”. (“D” stands for “summability domain of sequences”.{1a, 2a}. Accordingly, summability domains of series{1a} are compared, sets of sequences being considered preferably. (In German, the domain of a summability method is called its Wirkfeld{1b}, i.e. “action field”, “domain of efficiency”.) Thm. 5.2 asserts the methods C1 and A to be consistent{2b, 3a} (also called compatible{1c, 2b}) because any s ∈D(C1) ∩ D(A) (= D(C1)) is assigned the same value by C1, A. Our inclusion above is largely articulated “A is stronger than C1” {1c, 2c}. So to speak: “stronger in the weak sense”, as the wording would mutually apply to methods where D(M1) = D(M2). (Below, we will show A to be strictly stronger than C1 which, in {3b}, is called “stronger”.) Inclusion may{3a, 4a, 5} or may not{1c, 2c} comprise consistency; in any such case

5.3 Features of Cesàro and Abel means

99

“stronger plus weaker makes equivalent”. Among authors, there is quite some discord on notation and symbols. * * * To demonstrate that A is strictly stronger than C1, we are going to show: The test series ∞ Σk=0 (–1)k (k+1) is summable A yet cannot be summable C1. It is summable A as show its

Abel means ∞ Σk=0 (–1)k (k+1) xk =

1 (1 + x)2 , | x | < 1;

(1)

when starting for the right (but also see Appx. E-6), this is clear through Cauchy’s Thm. 3.4: k k ∞ ∞ ∞ ( Σ∞ (– x)i ) (Σj=0 (– x)j ) = Σk=0Σi=0 (– x)i (– x)k–i = Σk=0 (– x)k Σi=01, | x | < 1. i=0

If we assume the test series to be summable C1 (Def. 5.1(ii1) ), its partial sums sn must satisfy (n + 1) σn = s0 + ... + sn with σn → λ, say, thus 1 1 n sn = n[(n + 1) σn – n σn –1] → λ – λ = 0

and hence (sn – sn –1 )/n → 0, whereas sn – sn –1 = (–1)n(n + 1). – – To demonstrate D(A)\D(C1) ≠ Ø, there was no need to know sn explicitly. The calculation indicates: If (sn) is summable C1 then sn /n → 0. We note: Theorem 5.7. In order that the sequence sn is summable C1 it is necessary that sn /n → 0 . Colloquially, the condition says that, if | sn | goes to infinity at all, it must go less rapidly than the sequence n does. Thm. 5.7 is restrictive in that D(C1) only may house sequences whose terms behave adequately moderate, as concerns oscillation. Such a condition on the terms’ growth, a so-called order condition, furnishes a limitation theorem for the method in question. The limitation theorem Thm. 5.7 proves to be best possible. How could one disprove that there might be “more than that” necessary for (sn) to be summable C1? To open the question: Must | sn | go even less rapidly than an essentially smaller sequence, that is some (n/un ) with some unbounded sequence un > 0 ? Which is to say: Does such (un) exist to the effect that sn(n/un)–1 = unsn /n → 0 is necessary for any (sn) to be summable C1? Assuming this to be true, in {6b} a sequence (sn*) ∈D(C1) is constructed such that unsn* /n ↛ 0. (Application: sn/ln n → 0 is not necessary for sn to be summable C1 since otherwise we would get unsn/n → 0 with un = n /ln n to be necessary.) Therefore, Thm. 5.7 is best possible and being called the limitation theorem of summability C1. * * * “Efficiency” is not confined to comparing a domain of summability with convergence or domains with one another. A method may be judged by its application to some definite end. This might be analytic continuation.

100

5 On the verge of summability

The geometric series Σ zk, z∈ℂ, converges on the open unit disc d(0; 1) ≔ {z ∈ℂ : | z | < 1} and diverges elsewhere as zk is not a null sequence if | z | ≥ 1. According to Leibniz, the func1

∞

tion g0 : z ↦ Σk=0 zk, z ∈ d(0; 1), is being continued to g : z ↦ 1− z, z ∈d(0; 1) ∪ {−1}, through continuity, by virtue of his law of continuity (5.2.1) that is. To be more precise about continuity: When performing 1 ∞ ∞ Σk=0 (−1)k xk = Σk=0(−x)k = 1+x →

1 2 as x↑1,

Leibniz virtually applied Abel summability to Grandi’s series and made g0 continue to a continuous continuation − and even an analytic continuation, so far efficient at only one point. Likewise does C1 (see below). To what extent will our methods do the job on ℂ\{1}? The enterprise works on all the boundary of d(0; 1) except for z = 1. Due to Frobenius’ Thm. 5.2., we need only prove this for C1 (also do Appx. E-7). As the transformation ν

1

(sν) ↦ (σn) in Def. 5.1 is linear, the partial sums sν(z) = Σk=0 zk = 1− z (1 – zν+1) of the geometric series go into the sequence 1 1 1 1 z 1 – zn+1 n n σn(z) = 1− z [n+1 Σν=01 − n+1 Σν=0 zν+1] = 1− z − (1 – z)2 n + 1 , z ≠ 1, 1

with limit 1− z as n → ∞. − ∞

If | z | > 1, the geometric series refuses summability A since Σk=0 zkxk cannot converge for any x < 1 with 1/| z | < x, i.e. 1 < | z x |. Hence, by Thm. 5.2 again, there is no summability C1 either. − − * * * When it comes to analytic continuation of the geometric series, the Abel method cannot do better than the arithmetic means, they both yield the same “meager” result. In 1895, Emile Borel conceived a regular method stronger than A and much more efficient in that respect.{4b} Remark 1 in 5.2.4 shows how Abel’s weighted means are generated through the geometric series. Accordingly, Borel would do so by virtue of the exponential series: xn

Definition 5.3. Provided that the series Σ n! sn converge for x ∈ℝ then (sn) is summable to λ by the method B if ∞ x

n

∞ x

n

(Σn=0 n! sn ) (Σn=0 n! )–1 → λ as x → ∞. The method B is regular (cf. the proofs at Thm. 5.3.(i), Appxs. C- 6, C-15), e.g.). n

1

z

Borel’s means of sn(z) = Σν=0 zν = 1− z − 1− z zn read β(x; z) = e−x

n

x Σ∞ s (z) = n=0 n! n

1 z −x ∞ 1 n 1− z − 1− z e Σn=0 n! (x z) .

5.3 Features of Cesàro and Abel means

101

The series converges everywhere; for x → ∞ we note e−x |ex z | = e−x ex Re z = ex(Re z – 1) → 0 and thus β(x; z) → 1/(1 – z) on the half plane {z ∈ℂ: Re z < 1}. In the present respect, B is more efficient than C1 and A. On the other hand, C1 is efficient when it comes to inquire into the behavior of the geometric series on the boundary of its convergence. Borel’s B rounds up our ABC of summability. {1} {2} {3} {4} {5} {6}

5.3.2

ZELLER; BEEKMANN: {1a} 5. {1b} 4f. {1c} 4. BOOS: {2a} 12. {2b} 23. {2c} 22. HARDY: {3a} 65. {3b} 66. KNOPP: {4a} [1] 480, III; [2] no.263. {4b} [1] 488–490; [2] 265 (7.). KOREVAAR [2] 4. PEYERIMHOFF: {6a} 15. {6b} 4 f.

The C1 limit: a continuous functional on a Banach space

In the course of its history, analysis happened on various levels of progress. At 4.4 we again experienced convergence and limit from another point of view, following the intention to reveal structure, to abstract it from its classical background. We will take a look on how that works in summability. Recall and keep some denotation, with C ≡ D(C0) in particular. The supremum norm 4.4(1) on linear spaces of bounded sequences was marked || .. ||; in the case of D(C0) we will write it || .. ||0 for the moment. As in the past, we regard any sequence s = (s0, s1, … , sν , …) ∈ℝ∞ subject to the linear and reversible transformation 1

n

T : s ↦ σ , where σ = (σn), σn = n+1 Σν =0 s ν (n = 0 , 1 , ...). As there are sequences summable C1 and unbounded (Appx. E-8), the sup norm does not apply to D(C1) = { s : T s ∈C } in total. This linear space will be normed such as to render the mapping T : D(C1) → D(C0) continuous with respect to the norm || .. ||0 on the image space. It is easily achieved by defining || s ||1 ≔ ||T s||0 , s∈D(C1),

(1)

as shows ||T sk – T s||0 = ||T ( sk – s)||0 = || sk – s||1 . (For s ∈C, see the Remark below.) The linear functional L: s = (sn) ↦ lim sn , s∈C, was shown in 4.4 to be continuous. Joining the continuous mappings T and L, the C1 limit is being performed by the linear functional s ↦ T s ↦ L(T s) ∈ℝ, s∈D(C1), and thus proves to be continuous on the normed real space D(C1) since |L(T sk) – L(T s)| ≤ ||T sk – T s||0 according as at Thm. 4.8 (L). * * *

102

5 On the verge of summability

In a line with Thms. 4.5, 4.7, we note Theorem 5.8. The linear space D(C1) of real sequences limitable C1 is complete under the norm (1). Proof. If (sk) is a Cauchy sequence in D(C1), then ||T sk – T sk+p ||0 = ||T (sk – sk+p)||0 = || sk – sk+p ||1

(2)

proves T sk to be a Cauchy sequence in C. By Thm. 4.7, there exists σ ∈C such that ||T sk – σ ||0 → 0 ; this, with s = T –1σ ∈D(C1), infers || sk – s ||1 → 0 by virtue of || sk – s ||1 = ||T (sk – s)||0 = ||T sk – T s ||0 = ||T sk – σ ||0 . – –

(3)

We refer to what Appx. D-12 is about: The normed space C0 of all null sequences is a Banach space. Does this have for counterpart the normed space D(C1)0 ≔ { s ∈D(C1): T s∈C0} of all sequences summable C1 to zero? To this, we interpret (2) and (3) with a Cauchy sequence (sk) in D(C1)0. By (2), (T sk) is a Cauchy sequence in C0. Through the reference above, (3) holds with some σ ≕ T s ∈C0 ; that yields sk → s ∈D(C1)0. Remark. The functional (1) on D(C1), when applied to s ∈C, induces the functional s ↦ || s ||1 on C which compares to || s ||0 by 1

n

|| s ||1 = ||T s ||0 = sup | σn | ≤ sup ( n+1 Σk=0 | s ν | ) = sup | sn | = || s ||0, s∈C. There is no equality since

1 2

= || s ||1 < || s ||0 = 1 for s = (0,1,0,0, …). * * *

Now that C has become a subspace of D(C1) under the common norm || . ||1, is it worthwhile to ask on what terms they live together in D(C1), the divergent and the convergent sequences. We will only consider the space D(C1)0 introduced above. Adopting the index notation in 4.4(5), (6), we write T : x = (x1, x2, … ) ↦ ξ = (ξ1, ξ2, … ), where ξn =

1 n

Σnν=1 xν , n = 0 , 1 , … .

By Cauchy’s limit theorem, there is C0 ⊆ D(C1)0; this relation cannot be depicted by a Venn diagram. The following shows every divergent sequence in D(C1)0 to neighbor some null sequence at any distance. Here is how, namely by sectional convergence.{2} Let T x = ξ ∈C0. With ei = (δji : j = 1,2, … ) as in 4.4, we have k

k

k

|| x – Σi=1 ξiT –1ei ||1 ≔ ||T (x – Σi=1 ξi T –1ei ) ||0 = || ξ – Σi=1 ξi ei ||0. k

The term on the right converges to zero by 4.4(6); for the left we note Σi=1 ξi T –1ei ∈C0 to

hold by T –1ei = i (ei – ei+1) , i = 1,2, … .

* * *

5.3 Features of Cesàro and Abel means

103

Sequences are functions, summability gains a lot from functional analysis when it comes to developing concepts. Problems are unified, ready for strategy. It is like taking an aerial view on the issues – which still leaves enough to be discovered on the ground. (See {1}.) With C1 for an example, we get a first glimpse on facets of summability. On the one hand, this discipline participates in a comprehensive theory covering continuous linear functionals on a large variety of spaces. On the other, its applications yield plenty of contribution to classical problems, put them into adequal context. (For a few samples, see the sequel.) {1} {2}

BOOS. ZELLER; BEEKMANN 30, 40, 42.

5.3.3

An early triumph of arithmetic means: Fejér’s Theorem

In 5.3.1, we talked about aspects of efficiency. It may also pertain to “straightness of results”. The theory of Fourier series was longing for such ones. We noted quadratic means to be successful in that Euclidean norm (4.4) and distance 3.3.4(4) furnished the limit 3.3.4(9). As regards to pointwise converging there had been no answer in the affirmative other than Dirichlet’s theory. At 3.3.4 we sketched the situation; in 1873, Leopold Fejér{1a} marked it by a striking example: He constructed a trigonometric sine series which converges uniformly on [0, π] thus providing a continuous sum function – to the effect that its Fourier cosine series diverges at x = 0. Dirichlet had started out from a closed representation of the Fourier polynomials: the Dirichlet Integral. It was a life’s time later when Fejér took it up and conceived the Fejér integral. It resulted in Theorem 5.9 (Fejér’s Theorem).{1b, 2a} Let the 2π-periodic function f : [– π, π] → ℝ be integrable. If f (x ± 0) exist, then the Fourier series of f is summable C1 to 3.3.4(6), i.e. [ f (x – 0) + f (x + 0)]/2. If f is continuous on I = [a, b] ⊆ [– π, π] then the C1 means of the Fourier sums converge uniformly on I.{1c, 2b} – Cauchy’s Thm. 5.3 makes Fejér’s Theorem infer Thms. 3.1, 3.2. {1} {2}

FEJÉR: {1a} [2] 1 f. {1b} [1] 59. {1c} [1] 60. KNOPP: {2a} [1] 511–514; [2] nos. 280 f. {2b} [1] 514; [2] no.282 (2.).

5.3.4

On the scale of Cesàro means

When Hölder considered arithmetic means as a “method” he at once thought of iteration (5.2.2, 5.2.4). Given s = (s0 , s1 , ... ), then 1

n

hn [s]≔ sn, hn(p)[s] ≔ n + 1 Σν=0 hν(p–1)[s], p = 1,2, … ,{1a, 2a, 3a, 4a, 5a} (0)

defines the Hölder transform (hn(p)) of (sn ) and constitutes the Hölder method Hp of order p = 0,1, ... ; hence H1 ≡ C1. With p > 0, it is a matrix method under the p-fold reversible matrix product C1 ◦ C1 ◦ ... ◦ C1. Cauchy’s limit theorem Thm. 5.3 provides D(Hp) ⊆ D(Hp+1) and D(Hp) ≠ D(Hp+1), p = 0,1, ... .

104

5 On the verge of summability

It becomes increasingly difficult to make a Hölder transform explicit, i.e. expressed in the terms of the sequence transformed. Cesàro’s merit was to cure this handicap by methods C2, C3, …{1b, 2b, 6a} respectively equivalent to H2, H3, … . (Appx. E-9.) The idea is as follows. For s = (sn : n = 0,1, ... ), the Cesàro sums (1) (“summatorical sequences”{4a} ) and the Cesàro means (2) of order p = 0,1, … are defined by (0)

(p)

Sn [s] = sn , Sn [s] =

Σnν=0 Sν(p–1)[s],

(1)

(p)

(p)

σn

Sn [s] {1c, 2c, 3b, 4b, 5b} [s] = (p) , Sn [1]

(2)

where 1 ≔ (1,1, … ). Thus, by ratio, s is being compared to the paradigm 1 of a convergent sequence, starting out from convergence of σn(0)[s] = sn/1. The Cesàro sums (1) become explicit in (6) below. Here, for example, we expound the Cesàro means (2) in case of p = 3, formally or on the grounds of power series convergence when assumed for sufficiently small | x | (cf. {5c}, also for induction). The recursion in (1) follows from the Cauchy product (p–1) n n ∞ ∞ ∞ (Σ∞ xi) (Σk=0 Sk [s]xk) = Σn=0Σν=0 xn–ν(Sv(p–1)[s]xν) = Σn=0[ Σν=0 Sv(p–1)[s]]xn (3) i=0

=

(p) Σ∞ S [s]xn , n=0 n

p = 1,2, … ,

by identity of power series (cf. 3.2.4). When used with p = 1,2,3 successively, (3) makes 1

3

(3) ∞ (1 – x) (Σ∞ s xn) = Σn=0 Sn [s]xn . n=0 n

(4)

This, together with 1

1 d

1

∞ n+2 (1 – x) = 2 (dx ) (1 – x)–1 = 2! Σ∞ n (n –1) xn–2 = Σn=0 ( 2 ) xn, n=2 3

2

yields through Cauchy’s multiplication that 1

n n – ν + 2 n–ν ∞ i+2 ∞ ∞ (1 – x) (Σ∞ s xn) = [Σi=0( 2 ) xi][Σk=0 sk xk] = Σn=0Σν=0( 2 ) x (sν xν) (5) n=0 n 3

∞

n

= Σn=0[ Σν=0 (

n– ν +2 2

) sν]xn .

Identity of the power series on the right of (4) and (5) makes us arrive at case p = 3 of the following formulas with p = 1,2, … (observe 3.2.3(2)): (p)

Sn [s]= (p)

Σnν=0 (n – νp +– 1p – 1) sν = Σnν=0 (n – νn +– νp – 1) sν ,{2c, 3c, 5d} n+p

n+p

Sn [1]= ( p ) = ( n ) . {2c, 5d}

(6) (7)

5.3 Features of Cesàro and Abel means

105

The binomial coefficients on the very right of (6), (7) stay meaningful if the order p assumes non-integer values, ready to define the scale of Cesàro methods Cα with at least α > – 1. In the sequel, p will denote non-negative integers. Only α = – 1, – 2, ... were excluded from becoming Cesàro orders{5d}; see {4d} for α ≥ 0 and {1d, 3c} for α > – 1. (The Hölder methods could be conceived for even all real orders.{4c}) Eventually, Cesàro’s and Hölder’s method were proven equivalent for all orders α > – 1. Nevertheless, there is this minor defect: σn(p) → ∞ infers hn(p) → ∞ (p = 2,3, ...), the converse is not true.{1e} * * * Since Cauchy’s Thm. 5.3 provides strict inclusion of the Hölder methods Hp, the equivalence of Hp and Cp accounts for strict inclusion of the Cesàro ones, too. As to compare Hp and Cp with A correspondingly, it will be Cp that is considered. For D(Cp) ⊆ D(A) see {1f}. To show D(Cp) ≠ D(A) {1g} (p∈ℕ), ∞

we may refer to the order of growth for summability Cp when applied to the series (S) Σk=0ak the Abel means of which are d 1 ∞ ∞ Σk=0 ak xk ≡ Σk=0 (– 1)k (k + p) (k + p – 1) ... (k + 1) xk = (dx) 1 + x , p

| x | < 1.

Clearly, S is summable A (cf. 5.3.1(1)), but cannot be summable Cp. Otherwise, by virtue of the general limitation theorem in{1h, 2d}, its coefficients would be required to satisfy ak /k p → 0, contrary to k p ≤ (k + p) (k + p – 1) … (k + 1) = | ak |. – – On his hunt for π, Nīlakantha saw to accelerate convergence of power series. Euler likewise transformed series to this end. As for series with positive terms a meaning is given to saying that one series converges better or faster than the other, and the same with divergence.{2e} Certain types of such series form systems the members of which compare that way, thus providing an “order” of convergence respectively divergence. It is quite another kind of hierarchy that Cesàro conceived by his orders of summability. (A divergent, yet C1 summable series is called “une fois indéterminée” by Cesàro and D(Cp)\D(Cp–1) comprises the sequences p-fold indeterminate.{6b}) * * * A series Σak is convergent only if ak → 0, by the Cauchy criterion (in 4.1); another necessary 1 n condition reads n Σk=1kak → 0 (5.3.6 (2) below). Such requirements of series convergence have their counterparts with Cesàro summability: If the series is summable Cp, p = 1,2, … , then (i) the sequence ak is limitable Cp to 0 {2f}; 1

n

(ii) the sequence n Σk=1 ka k is limitable Cp to 0 {2g}.

106

5 On the verge of summability

{1}

HARDY: {1a} 94. {1b} 103, Thm.49. {1c} 96 f. {1d} 97. {1e} 107, Thm.54. {1f} 108, Th.55. {1g} 108 f. {1h} 101, Thm.46, k' = 0. KNOPP: {2a} [1] 481 f; [2] no.265/1. {2b} [1] 498; [2] no.269. {2c} [1] 483; [2] no.265/3. {2d} [1] 501 f; [2] no.271. {2e} [1] 287 f; [2] no.162. {2f} [1] 502, Satz 4; [2] no.272. {2g} [1] 502, Satz 5; [2] no.273. ZELLER; BEEKMANN: {3a} 107 bottom. {3b}104 (1). {3c} 108. PEYERIMHOFF: {4a} 8. {4b} 8 f. {4c} 22. {4d} 47 f. BOOS: {5a} 100 bottom. {5b} 102–104. {5c} 103 f. {5d} 104. CESÀRO: {6a} 119 f. {6b} 119.

{2}

{3} {4} {5} {6}

5.3.5

Cesàro’s Cauchy product

Abel’s contribution Thm. 5.4 to the Cauchy product was improved when Cesàro employed arithmetic means, that is Thm. 5.5 to begin with. The philosophy of summability is strongly supported when, to their full extent, the Cesàro means are applied to the issue. Let us recall and put in line the results on Cauchy’s product which, so far, did not resort explicitly to absolute convergence of factor series. ∞

∞

∞

Suppose (i) Σi=0 a i = A, (ii) Σk=0 b k = B and let (P) Σl=0 c l denote the (formal) Cauchy product of the series. This infers the following. [1] If P converges to C then AB = C. (Abel; Thm. 5.4.) [2] If P is summable C1 to C then AB = C. References:

Σ|a i xi | < ∞, Σ|b k xk | < ∞ (0 < x < 1) (3.2.4(4)); ∞ ∞ (Σ∞ a xi )(Σk=0bk xk) = Σi=0 cl xl (0 < x < 1) (Cauchy; Thm. 3.4); i=0 i ∞

∞

∞

AB = lim x↑1 [(Σi=0 ai xi) (Σk=0bk xk)] = lim x↑1 Σi=0 cl xl (Abel; Thm. 4.2.); lim x↑1

∞ Σi=0 cl xl

=

Σ∞ c i=0 l

(Frobenius; Thm. 5.2).

[3] P is summable C1 to AB (Cesàro; Thm. 5.5). In the very paper that started by [3]{1a}, Cesàro provided thorough extension to summability as defined in 5.3.4.{1b} What was proved there for positive integer orders{1c, 2a} would later {2a} become extended to non-integer ones and then resulted in ∞

∞

Theorem 5.10 (Cesàro). With α, β > – 1, let Σi=0 a i , Σk=0 b k be summable Cα, Cβ to A, B, respectively. Then, their Cauchy product is summable to AB by Cα + β + 1, at least{3a}, i.e. summable Cγ with some γ > – 1 such that γ ≤ α + β + 1.{2b, 3a}

For the proof on integer orders see {3b}. As to non-integer orders, cf. Appx. E-4 with the folding {2c}. {1} {2} {3}

CESÀRO: {1a} 115 f. {1b}119 f. {1c}120. HARDY: {2a} 245, Notes § 10.3. {2b} 228 f, Thm.164. {2c} 229 (10.3.1). KNOPP: {3a} [1] 507 "2."; [2] no.277/end. {3b} [1] 506, Satz 9; [2] no.277.

5.3 Features of Cesàro and Abel means

5.3.6

107

Inverse theorems, Tauber’s theorem

Limitation deals with what happens beyond convergence. Yet, keeping in touch with the roots is vital. There is a most fertile branch that serves this aim. The basic idea might be introduced as follows. n

There is more than a tautology when saying that the partial sums sn = Σk=0 a k of a series and their arithmetic means σn =

1 n +1

Σnk=0 s k differ by difference, namely by s n – σn. To

bridge it, we may apply Abel’s partial summation 4.2.3(2) and get

Σnk=0 1·s k = Σn–1 k=0 (k + 1) (s k – s k+1) + (n + 1) s n 1

n

= (n + 1) s n – Σk=1 ka k ,

n

s n – σn = n + 1 Σk=1 ka k .

(1) ∞

Hence, by Cauchy’s limit theorem (Thm. 5.3), convergence of the series Σk=0 a k requires the property 1 n

Σnk=1 ka k

→ 0,

(2)

as had been noticed by Leopold Kronecker already who in {1} wrote 1

ψn

Σnk=1 ψk a k

→ 0 , where 0 < ψ1 < ψ2 < … < ψn → ∞.{2a}

Through (1), the “Kronecker condition” (2) becomes the link between summability C1 and convergence. Altogether we have 1

[ n+1 Σnν=0 s ν → λ &

1 n

Σnk=1 ka k

→ 0 ] ⇐  s n → λ,

(3)

where “⇐” is due to Cauchy and Kronecker, whereas “” covers Theorem 5.11. A series summable C1 is convergent if it satisfies the Kronecker condition (2).{2b, 3a} By Cauchy’s Thm. 5.3, the property ka k → 0

(4)

serves to be sufficient for (2); it is not necessary (Appx. E-10). To illustrate the inclusions (cf. Fig. 5.1) D(C0) = D(C1) ∩ K, K ≔ {(sn) : D(C0) ⊃ D(C1) ∩ k,

1 n

Σnν=1 ν (s ν – s ν–1)

k ≔ {(sn) : n (s n – s n–1) → 0},

→ 0},

(5) (6)

(with (6) being strict) we may say: (5) proves (2) to be the sieve just that fine as is necessary and sufficient to retain no more and no less than C ≡ D(C0) when D(C1) is strained through, whereas (4) plays the role of a filter too coarse as to collect all the convergent sequences. * * *

108

5 On the verge of summability

Thm. 5.11 might be instrumental in substantiating convergence; this is a minor aspect. Our concern will be another view on “efficiency”. In 5.3.1, efficiency of C1 is regarded the property to transfer a given divergent series into a convergent one; the limitation theorem (Thm. 5.7) proved C1 ineffective for divergent sequences or series which oscillate “too wildly”. Now, the Kronecker condition specifies a class of series with respect to which C1 is to be called ineffective for another reason. These series behave “too tamely”; they are not summable C1 unless when convergent.{3b}

K D(C1)

k C Fig. 5.1. C1 Tauberian theorems visualized.

* * * Cauchy’s limit theorem and Thm. 5.11 deal with the transformation T : s ↦ σ, s∈D(C1), in contrary ways: they either result in convergence of the image σ =T s or in convergence of the –1

pre-image s = T σ. The former successively yields the chain of theorems (Ths) D(C0) ⊆ D(C1), D(C1) ⊆ D(C2), … (5.3.4). On the other hand, Thm. 5.11 accordingly extends to ∞

Theorem 5.12. Let p = 0,1,… . A series Σk=0 a k summable Cp+1 is summable Cp if 1 n p +1

K n(p)[t] → 0, where t = (tn), tn ≔ K n(0)[t]≔

Σnk=1 ka k ,

n

K n(p+1)[t]≔ Σk=1 K k (p)[t] (n ≥ 1).{3c}

(Here, (K n(p)[t]: n = 1,2, …) denotes the p-fold summatorical sequence of the Kronecker sums tn; cf. 5.3.4.) For obvious reasons, the inclusions in (Ths) and in Thm. 5.12 form theorems which respectively are named direct{4a} and inverse{4a, 4b}. In general, inverse theorems pertain to comparable methods in that they provide information about summability by the weaker method (5.3.1). This also makes an inverse theorem: Whenever 0 < α < β, then D(Cα) ⊇

D(Cβ) ∩ B holds with B the set of bounded sequences{5a }; joining with D(Cα) ⊆ D(Cβ), it infers D(Cα) ∩ B = D(Cβ) ∩ B{5b, 6}, a theorem which is both a direct and an inverse one. * * *

The career of inverse theorems started when the Austrian Alfred Tauber wrote one and only one paper, enough to render his name immortal in summability after Hardy had coined “Tauberism”{2c}. Those five pages of Tauber’s refer to Abel means alone. Surprisingly, Kronecker’s condition (2) turns out to play the same part in case of summability A as it does with the much weaker method C1. Tauber’s analogue to Thm. 5.11 states: Theorem 5.13. A series summable A is convergent if it satisfies the “Kronecker condition” (2).{7a}

5.3 Features of Cesàro and Abel means

109

The counterparts of (6), (5) with C1 being replaced by A use to be called, in this order, Tauber’s first {3d, 7a, 8a} and second{3e, 7b, 8b} theorem. Properties (2) and (4) are named Tauberian conditions for the methods C1 and A. The whole class of inverse theorems became denoted Tauberian theorems. Tauber’s inverse theorem was on Abel summability; the latter’s name will be involved in this context the other way. Like Cauchy’s limit theorem on arithmetic means, the one of Abel is a direct theorem for it draws inference from convergence of a series Σa k to a property of its ∞

Abel means Σk=0 a k xk ({8c}; 5.2.4, Remark 1). Due to this, direct theorems go by the name Abelian. {1} {2} {3} {4} {5} {6} {7} {8}

KRONECKER 980. KNOPP: {2a} [1] 131, Satz 3; [2] no.82. {2b} [1] 503, Satz 6; [2] no.274. {2c} [1] 503 f, footn.2; [2] no.274, footn.2. HARDY: {3a} 122, Thm.65, r = 0. {3b} 121. {3c} 122, Thm.65. {3d} 149, Thm.85. {3e} 150. {4a} ZELLER; BEEKMANN 55. {4b} KOREVAAR [2] 6. PEYERIMHOFF: {5a} 66, Theorem III.1. {5b} 66, Theorem III.2. BOOS: {6a} 176, Theorem 4.1.14, α = 0. {6b}177, Corollary 4.1.16. TAUBER: {7a} 274 (“B”). {7b} 273 (“A”). KOREVAAR [2]: {8a} 10 f. {8b} 11. {8c} 1(1.2).

Appendixes (of notes, proofs, exercises) *********************************** A ************************************ Appx. A-1 (section 1.1). (William Dampier was an explorer, an author and allegedly (?) a pirate around 1700, yet ...) Sir William Cecil Dampier, formerly Whetham, tells us that Zeno’s problem{3a} “could only be resolved completely” (cf. {3b}) after Cantor had detected the variety of infinities…{4a}. More twilight is shed where Dampier reasons about “the problem of infinity and that of continuity”: he considers them to be “essentially” the same, “for a continuous series must have an infinite number of terms”{4b}. ************************************************************************** Appx. A-2 (1.2). Exercise. Given any triangle, construct by ruler and compass the square equal in size, one way or the other. (First, think of a rectangle twice as large.) ************************************************************************** Appx. A-3 (1.2). Compare Fig. 1.1, i.e. {2a: [1] Fig. 46, [2] Fig. 72}, with {2d: [1] Fig. 63, [2] 96}, {6a: Fig. 4.3}. ************************************************************************** Appx. A-4 (1.3). Exercise. Given any tetrahedron, change it into an equal one that shows three right angles with a common vertex (like M in Fig. 1.1). (Apply Eudoxus’ result, twice.) ************************************************************************** Appx. A-5 (1.3). From Archimedes{2d, 3d}: The vertex P of the triangle ABP is as well the “vertex” of the parabolic segment SAB{2d, 3d) in that, among all points of the latter, the maximal distance to AB is assumed at the point P on the tangent T. Exercise: Show that, in Fig. 1 1

1.3, the point P( 2 | 4) is the vertex of SAB. ************************************************************************** Appx. A-6 (1.3). Like the arithmetic (i) sequences, a geometric (ii) and a harmonic (iii) sequence is defined through the respective mean sk in (i), (ii), (iii) of adjacent terms. With k = 1,2, … , we note (i) sk =

1 2

(sk–1 + sk+1), with c, d ≥ 0 hence sk = c k + d ;

(ii) sk = (sk–1 sk+1)½ where s0, s1 > 0, with c, q > 0 hence sk/sk–1 = q, sk = c qk; 1

1

1

1

(iii) s = 2 (s + s ) where s0, s1 > 0 (the reciprocal of the harmonic mean is the arithmek k–1 k+1 tic mean of its neighbors’ reciprocals).

112

Appendixes (of notes, proofs, exercises)

*********************************** B ************************************ Appx. B-1 (2.2). Comment on Fig. 2.3b. The two equal segments x within the square on the right qualify for being equal indivisibles in that their copies on the left, according to Cavalieri’s principle, are at equal distance to their triangles’ bases (or tops).{6, 7a} The argument given in {8a}, lines 5/6, is fallacious; it ignores, within four pages, Cavalieri’s counterexample {7b, 8b}. According to Cavalieri, pairs of triangles are equal in size on account of being entities of equal indivisibles. Next to this and in our words, equally high tetrahedrons have equal volumes if their bases are equal in area. *********************************** C ************************************ Appx. C-1 (3.2.3). On Newton’s interpolation of Pascal’s triangle. At {3g}, the column entrance of “Table 5” must read "2n" instead of “n”. The quotation {6a} as given at {3c} lacks thorough reference to the algebraic strategy{3h} by which Newton filled the gaps. (Useless: {1c}.) ************************************************************************** Appx. C-2 (3.2.4). Changing sum notation at 3.2.4(1). The range k ≤ j ≤ n of j is changed into the range 0 ≤ i ≔ j – k ≤ n − k of i; thereafter, j is ren sumed instead of i. Formal compensation: In Σj=k Tj replace T j by T j+k and lower the sumn– k

mation limits k and n by k each to result in Σj=0 T j+k. ************************************************************************** Appx. C-3 (3.2.4). A non-expanding Taylor series (according to Cauchy’s example). Cauchy employed f (x) = e−φ(x) with φ(x) = x−2, where we take φ(x) = | x |−1 (as also do some textbooks), for x ≠ 0 and f (0) ≔ 0. Since f (− x) = f (x) , we consider f (x) on [ 0,∞) only and start from f (x) = e−1/x, x ≥ 0. There, the derivative f (n)(x), n = 0,1, ... , is the product of f (x) 1

1

with a polynomial in x each since the derivative of such a polynomial is a polynomial in x 1

again. Now, t k/et → 0 as 0 < x = t → ∞, for all k = 0,1, ... , by l’Hospital’s rule. So, for all n = 0,1, ... , there holds f (n)(x)/x → 0 as x → 0 which, step by step, yields f (x) − f (0) 1 = f (x)· x → 0 as 0 < x → 0, hence: f '(0) = 0, x−0 f '(x) − f '(0) 1 3 = f (x)·( x ) → 0 as 0 < x → 0, hence: f "(0) = 0, x−0

and so forth. Thus, the Taylor series of f about x0 = 0 reads 0 + 0 x + 0 x2 + … , which represents the function f but trivially, that is at x = 0. (Exercise: Perform the induction where it applies.) **************************************************************************

Appendixes (of notes, proofs, exercises)

113

Appx. C-4 (3.2.4). Exercise. Prove that, in ℝ, an absolutely convergent series is convergent. ************************************************************************** Appx. C-5 (3.2.5). On the radius of convergence. (Reference: PENROSE [1] 80; [2] 126.) n

Exercise. What do the graphs of sn(x)≔ Σk=0 (−1)k x2k, n = 0,1, … , tell us about convergence and divergence of the sequence? Comment on the unmarked sketch Fig./Abb. 4.2 (at [2], the legend is misprinted in three places; also read x = 2 on p.125). ************************************************************************** ∞

1

1

Appx. C-6 (3.2.5). Proof of lim (1 + n )n = Σk=0 k! . 1

1

1

(Left hand: see 3.2.3(1),(2) for a polynomial in n. Right hand: k! ≤ (k −1)k , k ≥ 2; cf. 2.3(3).) We will show that the (positive) differences n

1

Dn≔ Σk=0 k! −

Σnk=0 ( nk ) (n1 )k

=

... (n−k +1) Σnk=1 k!1 (1 − n(n−1) ) n · n · ... · n ∞

ε

1

form a null sequence. Given ε > 0, with n1 such that Σk=n k! ≤ 2 and with n ≥ n1 we write 1 Dn =

Σ1 ≤ k < n k!1 [1 − (1 − 1n ) ·…· (1 − k n−1 )] + Σn ≤ k ≤ n 1

1

ε

1 ε k! ≤ s(n; n1) + 2 ,

ε

say. Thus, for some n2 ≥ n1, we get Dn ≤ 2 + 2 whenever n ≥ n2 . – Exercise: How does the proof extend to the result of 3.2.5(1) for all real x? ************************************************************************** Appx. C-7 (3.2.5). Euler: ii = e−(1/2 + 2k) π, k∈ℤ . On 3.2.5{8d}, p.491 line 2 from below, the formal procedure might as well suggest the following: − 1 = eiπ(eiπ)2k = eiπ(1+2k), hence i = eiπ (1/2 + k) which yields the wrong value for ii. (One happens to be better off with the formal procedure i = eiπ/2(eiπ)2k = eiπ(1/2 + 2k).) ************************************************************************** Appx. C-8 (3.2.6). On Euler’s first proof for

2

Σk∞= 1 k12 = π6 .

If a polynom of degree n has n real, possibly multiple, zeros xν ≠ 0 (ν = 1,…,n) then it allows for the decomposition a0 + a1 x2 + … + an xn = an (x − x1)(x − x2) … (x − xn) x

x

x

= (−1)n x1 x2 … xn · (1 − x ) (1 − x ) … (1 − x ). 1

2

n

114

Appendixes (of notes, proofs, exercises)

For reason of its zeros kπ, k∈ℤ\{0}, Euler considered the function f (x) =

sin x x2 x4 = 1 − x 3! + 5! − + … , f (0)≔ f (±0) ≠ 0,

(1)

to suggest, by analogy, the representation x π

f (x) = (1 + )(1 −

x x x x2 x2 ) · (1 + )(1 − ) · … = (1 − )·… 2 )(1 − 2π 2π (1π) (2π)2 π

≡ F1(x) · … · Fk (x) · … .

(2) ∞

When “figured out”, the infinite product formally yields Σj = 0 c2j x2j to be compared with (1). For what Euler was after, he determined the coefficient c2 in order to equal −1/3! in (1); this is how and why. Each factor Fk in (2) contributes to c2 x2 by the additive term tk(x) = 1 · … · 1 · (− ∞

= −[Σk = 1

x2 ) · 1 · 1 · … , resulting in c2 x2 = k2 π2

1 ] x2 and thus k2 π2

Σk∞= 1 tk(x)

1 ∞ 1 1 Σ 2 = 3! . π2 k = 1 k

************************************************************************** π

Appx. C-9 (3.3.4). Exercise. Show ∫ –π (cos m x)(sin n x) dx = 0 (m, n = 1,2, …) by verifying 4i (cos m x)(sin n x) = e i (m+n) x − e −i (m+n) x − [e i (m−n) x − e−i (m−n) x] = 2i [sin (m+n) x − sin (m−n) x]. ************************************************************************** Appx. C-10 (3.3.4). Provided that f (x) is integrable on some interval (−l, l ), its Fourier series is obtained through the Fourier series of φ(u) ≔ f (x) where x = (l/π)u, – π < u < π, namely π π α ∞ f ~ 20 + Σk=1 [αk cos k l x + βk sin k l x] in terms of the Fouries coefficients αn, βn of φ. Exercise: Let f (x), x∈(a, 3a), be integrable and 2a-periodic. Construct the Fourier series of f. ************************************************************************** Appx. C-11 (3.3.4). A note on 3.3.4{3e}. If different “trigonometric expansions” are called in question, this would contradict (7) unless being interpreted with reference to a period interval. **************************************************************************

Appendixes (of notes, proofs, exercises)

115

Appx. C-12 (3.4.1). (Concerning uniform convergence.) In order to prepare for term-by-term integration in (2), the integrand g(γρ(t)) γρ'(t) is repre∞

sented by Σk = 0 Tk (t; ρ), say, according to the Taylor expansion of g(z) that precedes (2). Exercise. Look for a series Σ ck (ρ) < ∞ such that |Tk (t; ρ) | ≤ ck (ρ), 0 ≤ t ≤ 2π. ************************************************************************** Appx. C-13 (3.4.1). Exercise. a)

What is the value of ∫γ[1| ρ] g(z) h(z) dz, ρ > 0 ?

b)

Find Laurent expansions of the function z (1 − z) about its poles.

1

************************************************************************** Appx. C-14 (3.4.2). Exercise (on the Cauchy product of the series in 3.4.2(9)). l

Confirm: Σi=0 ui υl – i = 0 for l = 1,2, … . ************************************************************************** Appx. C-15 (3.4.2). Mertens’ Theorem. (Also confer Appx. E-3.) n

ν

Let Un ≔ Σi=0 ui → U, Σ| ui | < ∞, Vν ≔ Σk=0 υk → V. The partial sums Cn of the Cauchy product take the form n

Cn = Σl=0

Σli=0 ui υl–i = Σni=0 ui Σnl=i υl–i = Σni=0 uiVn–i = Σnν=0 un–ν Vν

(0 ≤ i ≤ l ≤ n, 0 ≤ i ≕ n – ν ≤ n, 0 ≤ ν ≤ n; cf. Appx. C-2). So, n

Cn = Σν=0 un–ν (Vν – V ) + V Un → V U

if

Σnν=0 un–ν (Vν – V )

→ 0 as n → ∞. ε

∞

Write α ≔ Σi=0 |ui | (assume α > 0), rν ≔ Vν – V. Given ε > 0, let n0 be such that | rν | ≤ 2α for all ν ≥ n0; then (cf. the proofs of Thm. 5.3 and in Appx. C-6)

| Σnν=0 un–ν rν | ≤ | Σ0 ≤ ν < n0 un–ν rν| + Σn0 ≤ ν ≤ n | un–ν | | rν |

ε

ε

≤ 2 + α 2α = ε

with all sufficiently large n ≥ n0, since un–ν → 0 for n → ∞, ν = 0, … , n0. *********************************** D ************************************ Appx. D-1 (4.2.2). Exercise. Show that any function {1, 2} → ℝ fulfills (i), (ii) in Def. 4.1. ************************************************************************** Appx. D-2 (4.2.2). Exercise. It looks evident that the function 1/x is not uniformly continuous on the interval (0, 1). Prove it. **************************************************************************

116

Appendixes (of notes, proofs, exercises) ∞

Appx. D-3 (4.2.3). Exercise. Assume Thm. 4.2 to be insured in case of Σk=0 ak = 0. Complete the proof. ************************************************************************** Appx. D-4 (4.2.3). Exercise. Show by direct disprove (which is by employing but the definition of uniform convergence) that the sequence (1, x, x2, … ) does not converge uniformly on (0, 1). ************************************************************************** Appx. D-5 (4.2.3). Exercise. Prove or disprove that, on (0, 1), the function 1

1

(a) 1 – x , (b) 1 + x has a uniformly convergent Taylor series about x0 = 0. ************************************************************************** –1

Appx. D-6 (4.2.3). Exercise. Let fn(x) ≔ n x [1 +(n x)2] , 0 < x < 1 . Show: a)

lim x → ξ lim n → ∞ fn(x) = lim n → ∞ lim x → ξ fn(x), 0 < ξ < 1. 1

∫ 0 dx lim n → ∞ fn(x) b)

1

= lim n → ∞ ∫ 0 fn(x) dx. 1

The sequence fn does not converge uniformly. (Hint: fn ( n )?)

************************************************************************** Appx. D-7 (4.3). Euclid’s proof for √2 to be irrational. Assume c2 = 2 and c = p/q where p, q ∈ℕ do not share a divisor other than 1. The assumption (c2q2 =) 2q2 = p2 requires p = 2r with some r ∈ℕ, for otherwise p2 would be odd. For the same reason, q is even, too, since 2q2 = 4r2, q2 = 2r2; this would make 2 a common divisor of p and q. ************************************************************************** Appx. D-8 (4.3). An equivalence relation “∼” on whatever set of elements α, β, γ, … is thus defined: (i) α ∼ α; (ii) if α ∼ β then β ∼ α; (iii) [if α ∼ β and β ∼ γ] then α ∼ γ. Let [α]≔ {ξ: ξ ∼ α}. Exercise. Prove by distinct reference to (i), (ii), (iii): If and only if α ∼ β then [α] = [β]. ************************************************************************** Appx. D-9 (4.3). Given the structure (ℕ0,+), where ℕ0 = ℕ ∪{0}. Exercise. Re-interpret the construction of (ℚ+,·) so as to generate (ℤ,+). (How does Appx. D-8 apply here?) **************************************************************************

Appendixes (of notes, proofs, exercises)

117

Appx. D-10 (4.3). Exercise. Let the sequence of rational numbers cn in 4.3(3) generate the equivalence class C. Compute an element of C2. ************************************************************************** Appx. D-11 (4.4). Exercise. Show, by the ordinary Pythagorean theorem, how the norm ||(x, y, z)|| ≔ √ (x2 + y2 + z2) on the vector space ℝ3 makes the length of a segment in Euclid’s geometry. ************************************************************************** Appx. D-12 (4.4). Exercise (regarding (N ), (L) ). Ask yourself, not the books, why any norm satisfies | ||u|| – ||v|| | ≤ ||u – v||; | lim un | = lim |un| if lim un exists. ************************************************************************** Appx. D-13 (4.4). Exercise. Show: s1 e1 + s2 e2 + … = t1 e1 + t2 e2 + … holds only if s1 = t1 , s2 = t2 , … . *********************************** E ************************************ Appx. E-1 (5.2.2). The proof is heavy-going. In {1b} read ⎪an+k⎪ < 4ε (n+k). – Remark: n

The assumption σn≔ (n + 1)−1Σν=0 sν → λ readily infers, through sn = (n + 1) σn – n σn−1, that sn n + 1 an n = n σn − σn−1 → λ – λ = 0 and thus also n → 0.

************************************************************************** Appx. E-2 (5.2.2). Exercise. Thm. 5.3 being proven for λ = 0, prove it for any λ ∈ℂ . ************************************************************************** Appx. E-3 (5.2.3). Cauchy product: Iterated sum of resultants. (Cf. 3.4.2(2).) ν

ν

l

In Cν = Σl = 0 cl = Σl = 0 Σi = 0 ai bl – i the terms form an isosceles rectangular triangle with a0 bν , a1 bν –1 , … , aν b0 the hypotenuse. According to 0 ≤ i ≤ l ≤ ν, the leading index l makes i vary between 0 and l. With i the leading index, the double sum writes ν

ν

Cν = Σi = 0 ai Σl = i bl – i =

Σνi = 0 ai Bν – i

ν

or (by i = ν – k) Cν = Σk = 0 aν – k Bk .{3b, 5c}

(This compares to 3.4.2(2), as does the following.) From the latter we obtain, in a similar way,

Σnν = 0 Cν = Σνn= 0 Σνk = 0 aν – k Bk = Σ0 ≤ k ≤ ν ≤ n aν – k Bk = Σ nk = 0 Bk Σνn=k aν – k = =

Σ nk = 0 An – k Bk = Σni= 0 Ai Bn – i .

**************************************************************************

118

Appendixes (of notes, proofs, exercises)

Appx. E-4 (5.2.4). Exercise. a)

Show Σ kpxk < ∞ for all p = 1,2, … .

b)

Check Σk = 0 (–1)k (k + 1)(k + 2) ... (k + p) , p = 1,2, … , for being summable A.

∞

************************************************************************** Appx. E-5 (5.2.4). Total regularity of the methods C1, A; proof of Thm. 5.6. Thm. 5.6(i). Given P > 0, there is n(P) such that sν ≥ 2P for ν > n(P). So, with all sufficiently large n > n(P), we get

Σnν=0 sν ≥ Σ

n(P) < ν ≤ n

1

n(P)

n

1

sν ≥ (n – n(P)) 2P, hence n Σν=0 sν ≥ (1 – n ) 2P ≥ 2 2P = P. ∞

Thm. 5.6(ii). Assume, through (2), that α(x) ≔ (1– x) Σν=0 sν xν, x∈(ξ, 1). Given P > 0, there is n(P) such that sν ≥ 2P for ν ≥ n(P). Hence ∞

1

α(x) ≥ (1– x) 2P Σn(P) ≤ ν xν = (1– x) 2P xn(P) Σν=0 xν = 2P xn(P) ≥ 2P · 2 for all x ∈(ξ, 1) sufficiently close to 1. ************************************************************************** Appx. E-6 (5.3.1). Exercise. To achieve (1), the Cauchy product is not the easiest way. What kind of argument would Newton have used? ************************************************************************** Appx. E-7 (5.3.1). Exercise. Let z ≠ 1, | z | = 1. Show the direct way that the geometric 1

series is summable A to the value 1− z . ************************************************************************** Appx. E-8 (5.3.2). Exercise. Construct a sequence summable C1 that is unbounded. (Start out from its C1 transform.) ************************************************************************** Appx. E-9 (5.3.4). Exercise. Show: [D(C2)\D(C1)]∩ D(A) ≠ Ø. (Choose an appropriate H2 transform.) ************************************************************************** Appx. E-10 (5.3.5). Exercise. Show: The inclusion 5.3.6(6) is strict. **************************************************************************

Life Data Most data were adopted from WUẞING [1] Pythagoras of Samos Zeno of Elea Democritus of Abdera Plato Eudoxus of Knidos Aristotle Euclid of Megara Archimedes of Syracuse Apollonius of Perga “Fibonacci”, Leonardo of Pisa Roger Bacon Thomas Bradwardine Nicole d’Oresme (Oresmus) Nīlakantha Nicolaus Copernicus Michael Stifel John Napier (Neper) Jost Bürgi Henry Briggs Galileo Galilei Johannes Kepler Grégoire de Saint-Vincent Thomas Hobbes Bonaventura Cavalieri Pierre de Fermat Evangelista Torricelli John Wallis Nicolaus Mercator Blaise Pascal Christiaan Huygens James Gregory Isaac Newton Gottfried Wilhelm Leibniz Jacob Bernoulli Pierre Varignon Abraham de Moivre Johann Bernoulli Guido Grandi

≈ 569–475 BC ≈ 495–430 BC 460–371 BC 427–347 BC ≈ 408–355 BC 384–322 BC ≈ 360–290 BC ≈ 287–212 BC ≈ 262–190 BC 1170– ≈ 1240 ≈ 1214–1294 ≈ 1290–1349 1323–1382 15th/16th century 1473–1543 1487–1567 1550–1617 1552–1632 1556–1631 1564–1642 1571–1630 1584–1667 1588–1679 1598–1647 1607–1665 1608–1647 1616–1703 1620–1687 1623–1662 1629–1695 1638–1675 1643–1727 1646–1716 1654–1705 1654–1722 1667–1764 1667–1748 1671–1742

120 Brook Taylor Nicolaus Bernoulli Daniel Bernoulli Leonhard Euler Jean-Baptiste le Rond d’Alembert Joseph-Louis Lagrange Pierre-Simon Laplace Jean Baptiste Joseph de Fourier Carl Friedrich Gauß (Gauss) Bernhard [Bernardus] Bolzano Augustin Louis Cauchy Niels Henrik Abel J. Peter Gustav Lejeune Dirichlet Pierre Alphonse Laurent Karl Th.W. Weierstraß Heinrich Eduard Heine Leopold Kronecker G. F. Bernhard Riemann J. W. Richard Dedekind Franz Carl Josef Mertens Georg F. L. Cantor Friedrich Ludwig G. Frege Georg Ferdinand Frobenius Giuseppe Peano Otto Hölder Ernesto Cesàro David Hilbert Alfred Tauber Emile Borel Sir Bertrand A.W. Russell Henri L. Lebesgue Godefrey Harold Hardy György Pόlya Stefan Banach Kurt Gödel Nicolas Bourbaki

Life Data 1685–1731 1687–1759 1700–1782 1707–1783 1717–1783 1736–1813 1749–1827 1768–1830 1777–1855 1781–1848 1789–1857 1802–1829 1805–1859 1813–1854 1815–1897 1821–1888 1823–1891 1826–1866 1831–1916 1840–1927 1845–1918 1848–1925 1849–1917 1858–1932 1859–1937 1859–1906 1862–1943 1866–1942 1871–1956 1872–1970 1875–1941 1877–1947 1887–1985 1892–1945 1906–1978 1935– ∞

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Index A Abel, method of ~ means 97 Abel, Niels Henrik 31, 72, 89, 120 Abel’s limit theorem 73 partial summation 74, 96 Abelian theorem 109 actual infinite, the ~ 2 Alembert, Jean-Baptiste le Rond d’~ 40, 49, 120 analysis algebraical ~ 20, 42 arithmetization of ~ 68 Fourier ~ 54 functional ~ 86 of the infinite 5 analytic continuation 57, 100 function 56 ansatz by trigonometric series 49 power series ~ (for diff. equations) 35 Apollonius of Perga 10, 27, 119 Archimedean axiom 7 Archimedes 6, 9, 119 The Method, by ~ 9 arcustangent series 23, 72 area of the circle 5, 30 parabola (segment) 6, 14 Aristotle 2, 119 arithmetic ~ progression 28 means 96 arithmetization of analysis 68 B Bacon, Roger 11, 119 Banach space 86 Banach, Stefan 86, 120 base of logarithms 38 Bernoulli, Daniel 49, 92, 120

Bernoulli, Jacob 17, 91, 119 Bernoulli, Johann 17, 40, 119 Bernoulli, Nicolaus 44, 120 binomial 29 ~ coefficient 29, 31 ~ series/expansion 31 ~ theorem 29 Bolzano, Bernhard 55, 68, 120 Borel, Emile 100, 120 boundary condition 49 Bourbaki, Nicolas 55, 120 Bradwardine, Thomas 119 Briggs, Henry 27, 119 Bürgi, Jost 26, 119 C Cajori, Forian 27 Calculus (of Newton/Leibniz) 29 Cantor, Georg 80, 120 Cantor’s model for the real numbers 81 Cauchy criterion 82 postulate 42, 68 series/sequence 68, 82 Cauchy, Augustin Louis 55, 67, 120 Cauchy’s calculus of residues 59 integral formula 59 integral/main theorem 56 Cauchy’s product formula Abel’s theorem on ~ 94 Cesàro’s theorem on ~ 95, 106 Mertens’ theorem on ~ 115 Cavalieri, Bonaventura 4, 13, 119 Cavalieri’s indivisibles/entities 13 principle 4, 14 Cesàro means of natural orders p 104 noninteger orders α 105 order one 96 Cesàro sums 104 Cesàro, Ernesto 64, 90, 120

128 circle of convergence 34 quadrature of the ~ 5 closed evaluation of a series 43 form of the Fourier polynomials 103 path of integration 56 simple ~ path 57 coefficient binomial ~ 29 Fourier ~ 51, 52 Laurent ~ 58 Taylor ~ 32 Colson, John 30 complete normed space 78, 84, 85, 86, 102 completeness property in the terms of Cantor 82 Dedekind 81 complex numbers 55 condition boundary ~ 49 initial ~ 49 Kronecker convergence ~ 107 necessary ~ 77 congruent, dissectionally ~ 5 consistent methods of summability 98 continuous globally ~ 70 piecewise ~ 52 pointwise ~ 70 uniformly ~ 71 convergence ~ to a number 68 ~ in square (or quadratric) mean 54 circle/interval/radius of ~ 33 sectional ~ 87, 102 uniform ~ 75 convolution 61 Copernicus, Nicolaus 119 Cotes, Roger 40 criterion, Cauchy ~ 82 D Davis, Philip 53 decimal fraction 20, 68, 82 Dedekind cut 81 Dedekind, Richard 80, 120 Dedekind’s model for the real numbers 81

Index Democritus 1, 15, 119 differentiate, to ~ term by term 34 differentiation complex ~ 56 early ~ of the tangent function 24 Dirichlet, Peter Gustav 50, 54, 120 discontinuity, jump ~ 52 disk, open/pierced ~ 56 dissectionally congruent 5 distance, Euclidean/norm defined ~ 52 domain of summability 98 open ~ of a function 55 E efficiency of summability 90, 100 entities, Cavalieri’s ~ 14 epsilontics 8, 69 equivalence relation 116 equivalent methods of summability 99 Euclid 2, 9, 119 Euclidean geometry 4 norm/distance 52, 84 Eudoxus 3, 9, 119 Euler, Leonhard 36, 46, 49, 120 Euler’s equation 39 number e 36 valuation of series 44 Euler-Fourier formulas 50 evaluation, closed ~ of a series 43 exhaustion, method of ~ 4 expansion binomial ~ 29, 31 Fourier ~ 48, 51 geometrical ~ of fractions 20 power series ~ 34 Taylor ~ 32, 34 exponent first mention of ~ 26 imaginary ~ 40 exponential series/function 37 F Fejér’s theorem 103 Fermat, Pierre de 15, 119 Fibonacci [Pisanus] 11, 119 finite expressions, Euler’s ~ 44

Index formal series 37, 43 Fourier analysis 54 closed form of the ~ polynomials 103 coefficient 51, 52 series/expansion 48, 51 Fourier, Jean B. Joseph de 46, 120 fractional binomial powers 29 Frege, Gottlob 83, 120 Frobenius, Georg 92, 120 function analytic ~ 56 exponential ~ 37 multi-valued ~ 41 regular ~ 46 sawtooth ~ 47 transcendental ~ 41 functional 86 ~ analysis 86 linear/continuous ~ 101 fundamental/Cauchy sequence 68 G Galilei, Galileo 119 Gauß, Carl Friedrich 55, 120 general term of a series 43 geometric mean 7 progression 7, 8, 111 series 20, 21, 25, 100 Gödel, Kurt 83, 120 Grandi, Guido 43, 119 Grandi’s series 43, 47, 90 Gregory, James 21, 31, 69, 119 Gudermann, Christof 77 H Hardy, Godefrey Harold 42, 64, 76, 120 harmonic mean 12, 111 series 12 triangle 17 harmonics 49, 78 Heine, Eduard 71, 120 Hersh, Reuben 53, 60 heuristics 6, 9 Hilbert, David 83, 120 Hindu mathematics 26 Hobbes, Thomas 11, 119

129 Hölder means 103 Hölder, Otto 93, 103, 120 Huygens, Christiaan 16, 119 hyperbola of Apollonius 27 hyperbolic logarithm 37 I identity theorem for power series 34 imaginary exponent 40 incommensurable 78 incompleteness theorem, by Gödel 83 indivisible, the ~ 1 indivisibles Cavalieri’s ~ 13 equal ~ 15, 112 infinite, the actual/potential ~ 2 infinitesimals 10 infinity 1 initial condition 49 integral curvilinear ~ 56 Dirichlet ~ 103 Fejér ~ 103 Riemann ~ 53, 71 integral formula, Cauchy’s ~ 59 integral theorem, Cauchy’s ~ 56 integrate, to ~ term by term 57 interval of convergence 33 period ~ 50 K Kepler, Johannes 27, 119 Kronecker, Leopold 107, 120 L Lagrange, Joseph-Louis 31, 120 Laplace, Pierre-Simon 35, 120 Laurent coefficient/series 58 Laurent, Pierre Alphonse 58, 120 Lebesgue, Henri 53, 120 Leibniz, Gottfried Wilhelm 16, 91, 119 Lejeune Dirichlet See Dirichlet Lemma of Archimedes 7 limit notion of ~ 69, 86 point 69 limit theorem Abel’s ~ 73 Cauchy’s ~ 93

130 limitable/summable See summable/limitable limitation by the order of growth 99 limitation theorem for C1 99 for Cp 105 logarithm hyperbolic ~ 27, 37 natural ~ 27, 38, 41 long division of polynomials 21 loss of certainty 78, 83 M MacLaurin, Colin 30 matrix method of summability 97 mean geometric ~ 7 harmonic ~ 12, 111 square or quadratic ~ 52 weighted ~ 97 mean proportional 7 means arithmetic ~ 103 general Cesàro ~ 105 Hölder ~ 103 Hölder/Cesàro ~ of natural orders p 104 Mengoli, Pietro 17, 28 Mercator, Nicolaus 28, 119 Mertens’ theorem 64 method of power series (for diff. equations) 35 method of summability a ~ stronger/strictly stronger than a ~ 98 matrix ~ 97 regular/permanent ~ 98 Method, The ~, by Archimedes 9 model for real numbers Cantor’s ~ 82 Dedekind’s ~ 80 Moivre, Abraham de 39, 119 N Napier, John 26, 119 natural logarithm 38 necessary condition 77 Newton, Isaac 21, 29, 119 Nīlakantha 24, 119

Index norm Euclidean ~ 52 maximum/supremum ~ 84 postulates 84 null sequence 69 number all is ~ (Pythagoras) 78 calculate an irrational ~ 79 complex ~ 33, 40 Euler’s ~ e 36, 37 irrational ~ 78, 116 over-infinite ~ 44 pi [π] 25 rational ~ 79 transcendental ~ 39 numbers complex ~ 40, 56 figurate ~ 16 Greek ~ 3 Indian/Arabic ~ 26 triangular ~ 16 O open disk/domain 56 order of growth 99 limitation by the ~ 99, 105 Oresme/Oresmus, Nicolas 11, 119 orthogonal pair of functions 51, 114 orthogonal series 51, 54 orthonormal system of functions 51 Oxford school 11 P parabolas, higher ~ 13 partial fractions 16 summation, Abel’s ~ 74, 96 partition by classes 80 Pascal, Blaise 15, 25, 29, 71, 119 Pascal’s triangle 30 path of integration 56 simple closed ~ 57 Peano, Giuseppe 83, 120 Penrose, Roger 45 permanent method of summability 98 piecewise continuous 52 Plato 5, 119 platonist 83

Index polynomial Taylor ~ 32 trigonometric ~ 52 potential infinite, the ~ 2 power series 12, 22 ~ expansion 34 ~ method/ansatz (for diff. equations) 35 identity theorem for ~ 34 prism, trisected ~ 3 product of series, Cauchy’s ~ 61 progression arithmetic ~ 28, 111 exponential ~ 28 geometric ~ 7, 11 pyramid, Nile ~ 3 Pythagoras 119 Pόlya, György 22, 120 Q quadrature of the circle 5, 23 parabola (segment) 6 R rational number 79 real numbers Cantor’s model for the ~ 81 Dedekind’s model for the ~ 81 recurrent series 91 regular function 46 method of summability 90, 98 totally ~ method of summability 118 residue calculus 59 resultant 61, 117 Riemann, Bernhard 41, 53, 64, 120 Rudin, Walter 46 Russell, Bertrand 81, 120 S Saint-Vincent, Grégoire de 1, 8, 27, 119 Sarasa, Antonio de 27 sawtooth function 47 sectional convergence 87, 102 sequence Cauchy ~, fundamental ~ 68, 81 null ~ 69 summatorical ~ 104 series arcustangent ~ 23, 72 binomial ~ 31

131 Cauchy ~ 68 closed evaluation of a ~ 43 divergent ~ 48, 89 Euler’s algebraical view of ~ 42 exponential ~ 37 formal ~ 37, 43 Fourier ~ 48, 51 general term of a ~ 43 geometric ~ 20, 21, 25, 100 Grandi’s ~ 43, 47, 90 harmonic ~ 12 infinite ~ 4, 13, 20 Laurent ~ 58 Leibniz’ ~ 23 logarithmic ~ 28, 38 Mercator’s ~ 28, 38 orthogonal ~ 51, 54 power ~ 22 rearrangement of a ~ 64 recurrent ~ 92 Taylor ~ 32 Taylor’s ~ 29, 32 telescope ~ 16 trigonometric ~ 46, 51 simplex 2 singularity 56 (isolated) essential ~ 59 pole ~ 58 sophists 1 space Banach ~ 86 complete normed ~ 84, 85, 102 normed ~ 84 square mean 52 Stifel, Michael 26, 29, 119 Stifel’s law 30 Struik, Dirk 53 summability, theory of ~ 90, 97 summable/limitable A 97 C1 96 Cα 105 Hp, Cp 104 supremum norm (sup norm) 84 Swineshead [Suiseth], Richard 12 T Tauber, Alfred 108, 120 Tauberian theorem 109

132 Taylor coefficient/series 32 Taylor expansion of arcustangent 23, 72 natural logarithm 72 sine, cosine 39 Taylor expansion, definition of ~ 32 Taylor, Brook 32, 120 Taylor’s series 29, 32 telescope series 16 term-by-term differentiation 34 integration 57 tetrahedron 3 theorem Abelian ~ 109 binomial ~ 29 Fejér’s ~ 103 limitation ~. See limitation theorem Mertens’ ~ 64 of incompleteness, by Gödel 83 Tauberian ~ 109 Theory of Functions 55, 77 Torricelli, Evangelista 15, 119

Index triangle harmonic ~ 17 Pascal’s ~ 30 trigonometric polynomial 52 series 51 U uniform continuity 71 convergence 75 V Varignon, Pierre 91, 119 vibrating string 49 volume of the tetrahedron/pyramid 3 W Wallis, John 30, 44, 119 Weierstraß, Karl 69, 73, 120 Wolff, Christian 91 Z Zeno 1, 69, 119