Amazing and Aesthetic Aspects of Analysis

508 Pages·2007·3.48 MB·English

Amazing and Aesthetic
Aspects of Analysis:
On the incredible infinite
(A Course in Undergraduate Analysis, Fall 2006)
π2 1 1 1 1
= + + + +
6 12 22 32 42 ···
22 32 52 72 112
=
22 1 · 32 1 · 52 1 · 72 1 · 112 1 ···
− − − − −
1
=
14
02 + 12
− 24
12 + 22
− 34
22 + 32
− 44
32 + 42
− 42 + 52 ...
−
Paul Loya
(This book is free and may not be sold. Please email
[email protected] to report errors or give criticisms)

Contents
Preface i
Acknowledgement iii
Some of the most beautiful formulæ in the world v
A word to the student vii
Part 1. Some standard curriculum 1
Chapter 1. Sets, functions, and proofs 3
1.1. The algebra of sets and the language of mathematics 4
1.2. Set theory and mathematical statements 11
1.3. What are functions? 15
Chapter 2. Numbers, numbers, and more numbers 21
2.1. The natural numbers 22
2.2. The principle of mathematical induction 27
2.3. The integers 35
2.4. Primes and the fundamental theorem of arithmetic 41
2.5. Decimal representations of integers 49
2.6. Real numbers: Rational and “mostly” irrational 53
2.7. The completeness axiom of R and its consequences 63
2.8. m-dimensional Euclidean space 72
2.9. The complex number system 79
2.10. Cardinality and “most” real numbers are transcendental 83
Chapter 3. Infinite sequences of real and complex numbers 93
3.1. Convergence and ε-N arguments for limits of sequences 94
3.2. A potpourri of limit properties for sequences 102
3.3. The monotone criteria, the Bolzano-Weierstrass theorem, and e 111
3.4. Completeness and the Cauchy criteria for convergence 117
3.5. Baby infinite series 123
3.6. Absolute convergence and a potpourri of convergence tests 131
3.7. Tannery’s theorem, the exponential function, and the number e 138
3.8. Decimals and “most” numbers are transcendental ´a la Cantor 146
Chapter 4. Limits, continuity, and elementary functions 153
4.1. Convergence and ε-δ arguments for limits of functions 154
4.2. A potpourri of limit properties for functions 160
4.3. Continuity, Thomae’s function, and Volterra’s theorem 166
iii
iv CONTENTS
4.4. Compactness, connectedness, and continuous functions 172
4.5. Monotone functions and their inverses 182
4.6. Exponentials, logs, Euler and Mascheroni, and the ζ-function 187
4.7. The trig functions, the number π, and which is larger, πe or eπ? 198
4.8. F Three proofs of the fundamental theorem of algebra (FTA) 210
4.9. The inverse trigonometric functions and the complex logarithm 217
4.10. F The amazing π and its computations from ancient times 226
Chapter 5. Some of the most beautiful formulæ in the world 237
5.1. F Euler, Wallis, and Vi`ete 240
5.2. F Euler, Gregory, Leibniz, and Madhava 249
5.3. F Euler’s formula for ζ(2k) 258
Part 2. Extracurricular activities 267
Chapter 6. Advanced theory of infinite series 269
6.1. Summation by parts, bounded variation, and alternating series 271
6.2. Liminfs/sups, ratio/roots, and power series 279
6.3. A potpourri of ratio-type tests and “big ” notation 290
O
6.4. Some pretty powerful properties of power series 295
6.5. Double sequences, double series, and a ζ-function identity 300
6.6. Rearrangements and multiplication of power series 311
6.7. F Proofs that 1/p diverges 320
6.8. Composition of power series and Bernoulli and Euler numbers 325
P
6.9. The logarithmic, binomial, arctangent series, and γ 332
6.10. F π, Euler, Fibonacci, Leibniz, Madhava, and Machin 340
6.11. F Another proof that π2/6= ∞ 1/n2 (The Basel problem) 344
n=1
Chapter 7. More on the infinite: ProduPcts and partial fractions 349
7.1. Introduction to infinite products 350
7.2. Absolute convergence for infinite products 355
7.3. Euler, Tannery, and Wallis: Product expansions galore 359
7.4. Partial fraction expansions of the trigonometric functions 366
7.5. F More proofs that π2/6= ∞ 1/n2 370
n=1
7.6. F Riemann’s remarkable ζ-function, probability, and π2/6 373
P
7.7. F Some of the most beautiful formulæ in the world IV 382
Chapter 8. Infinite continued fractions 389
8.1. Introduction to continued fractions 390
8.2. F Some of the most beautiful formulæ in the world V 394
8.3. Recurrence relations, Diophantus’ tomb, and shipwrecked sailors 403
8.4. Convergence theorems for infinite continued fractions 411
8.5. Diophantine approximations and the mystery of π solved! 422
8.6. F Continued fractions and calendars, and math and music 433
8.7. The elementary functions and the irrationality of ep/q 437
8.8. Quadratic irrationals and periodic continued fractions 446
8.9. Archimedes’ crazy cattle conundrum and diophantine equations 456
8.10. Epilogue: Transcendental numbers, π, e, and where’s calculus? 464
Bibliography 471
CONTENTS v
Index 481

Preface
I have truly enjoyed writing this book. Admittedly, some of the writing is
too overdone (e.g. overdoing alliteration at times), but what can I say, I was
having fun. The “starred” sections of the book are meant to be “just for fun”
and don’t interfere with other sections (besides perhaps other starred sections).
Most of the quotes that you’ll find in these pages are taken from the website
http://www-gap.dcs.st-and.ac.uk/~history/Quotations/
This is a first draft, so please email me any errors, suggestions, comments etc.
about the book to
[email protected].
The overarching goals of this textbook are similar to any advanced math text-
book, regardless of the subject:
Goals of this textbook. The student will be able to ...
comprehend and write mathematical reasonings and proofs.
•
wield the language of mathematics in a precise and effective manner.
•
statethefundamentalideas,axioms,definitions,andtheoremsuponwhichreal
•
analysis is built and flourishes.
articulate the need for abstraction and the development of mathematical tools
•
and techniques in a general setting.
The objectives of this book make up the framework of how these goals will be
accomplished, and more or less follow the chapter headings:
Objectives of this textbook. The student will be able to ...
identify the interconnections between set theory and mathematical statements
•
and proofs.
state the fundamental axioms of the natural, integer, and real number systems
•
and how the completeness axiom of the real number system distinguishes this
system from the rational system in a powerful way.
apply the rigorous ε-N definition of convergence for sequences and series and
•
recognize monotone and Cauchy sequences.
apply the rigorous ε-δ definition of limits for functions and continuity and the
•
fundamental theorems of continuous functions.
determine the convergence and properties of an infinite series, product, or con-
•
tinued fraction using various tests.
identifyseries,product,andcontinuedfractionformulæforthevariouselemen-
•
tary functions and constants.
I’d like to thank Brett Bernstein for looking over the notes and gave many
valuable suggestions.
Finally, some last words about my book. This not a history book (but we
try to talk history throughout this book) and this not a “little” book like Herbert
Westren Turnbull’s book The Great Mathematicians, but like Turnbull, I do hope
i
ii PREFACE
If this little book perhaps may bring to some, whose acquain-
tance with mathematics is full of toil and drudgery, a knowledge
of those great spirits who have found in it an inspiration and
delight, the story has not been told in vain. There is a largeness
about mathematics that transcends race and time: mathematics
may humbly help in the market-place, but it also reaches to the
stars. To one, mathematics is a game (but what a game!) and
to another it is the handmaiden of theology. The greatest math-
ematics has the simplicity and inevitableness of supreme poetry
and music, standing on the borderland of all that is wonderful
in Science, and all that is beautiful in Art. Mathematics trans-
figures the fortuitous concourse of atoms into the tracery of the
finger of God.
Herbert Westren Turnbull (1885–1961). Quoted from [225, p.
141].
Paul Loya
Binghamton University, Vestal Parkway, Binghamton, NY 13902
[email protected]
Soli Deo Gloria
Acknowledgement
To Jesus, my Lord, Savior and Friend.
iii