A quantity can be made smaller and smaller without it ever vanishing. This fact has profound consequences for science, technology, and even the way we think about numbers. In this book, we will explore this idea by moving at an easy pace through an account of elementary real analysis and, in particular, will focus on numbers, sequences, and series.
Almost all textbooks on introductory analysis assume some background in calculus. This book doesn't and, instead, the emphasis is on the application of analysis to number theory. The book is split into two parts. Part 1 follows a standard university course on analysis and each chapter closes with a set of exercises. Here, numbers, inequalities, convergence of sequences, and infinite series are all covered. Part 2 contains a selection of more unusual topics that aren't usually found in books of this type. It includes proofs of the irrationality of e and π, continued fractions, an introduction to the Riemann zeta function, Cantor's theory of the infinite, and Dedekind cuts. There is also a survey of what analysis can do for the calculus and a brief history of the subject.
A lot of material found in a standard university course on "real analysis" is covered and most of the mathematics is written in standard theorem-proof style. However, more details are given than is usually the case to help readers who find this style daunting. Both set theory and proof by induction are avoided in the interests of making the book accessible to a wider readership, but both of these topics are the subjects of appendices for those who are interested in them. And unlike most university texts at this level, topics that have featured in popular science books, such as the Riemann hypothesis, are introduced here. As a result, this book occupies a unique position between a popular mathematics book and a first year college or university text, and offers a relaxed introduction to a fascinating and important branch of mathematics.
Author(s): David Applebaum
Edition: 1
Publisher: Oxford University Press, USA
Year: 2012
Language: English
Pages: 256
Tags: Математика;Математический анализ;
Cover......Page 1
Contents......Page 14
PART I: APPROACHING LIMITS......Page 18
1.1 Natural Numbers......Page 19
1.2 Prime Numbers......Page 20
1.3 The Integers......Page 26
1.4 Exercises for Chapter 1......Page 29
2.1 The Rational Numbers......Page 31
2.2 Irrational Numbers......Page 34
2.3 The Real Numbers......Page 40
2.4 A First Look at Infinity......Page 41
2.5 Exercises for Chapter 2......Page 43
3.1 Greater or Less?......Page 45
3.2 Intervals......Page 50
3.3 The Modulus of a Number......Page 51
3.5 The Theorem of the Means......Page 55
3.6 Getting Closer......Page 58
3.7 Exercises for Chapter 3......Page 59
4.1 Limits......Page 62
4.2 Bounded Sequences......Page 71
4.3 The Algebra of Limits......Page 73
4.4 Fibonacci Numbers and the Golden Section......Page 76
4.5 Exercises for Chapter 4......Page 79
5.1 Bounded Sequences Revisited......Page 82
5.2 Monotone Sequences......Page 86
5.3 An Old Friend Returns......Page 88
5.4 Finding Square Roots......Page 90
5.5 Exercises for Chapter 5......Page 92
6.1 What are Series?......Page 95
6.2 The Sigma Notation......Page 96
6.3 Convergence of Series......Page 99
6.4 Nonnegative Series......Page 101
6.5 The Comparison Test......Page 105
6.6 Geometric Series......Page 109
6.7 The Ratio Test......Page 112
6.8 General Infinite Series......Page 115
6.9 Conditional Convergence......Page 116
6.10 Regrouping and Rearrangements......Page 119
6.11 Real Numbers and Decimal Expansions......Page 121
6.12 Exercises for Chapter 6......Page 122
PART II: EXPLORING LIMITS......Page 126
7.1 The Number e......Page 127
7.2 The Number π......Page 135
7.3 The Number γ......Page 140
8.1 Convergence of Infinite Products......Page 143
8.2 Infinite Products and Prime Numbers......Page 147
8.3 Diversion – Complex Numbers and the Riemann Hypothesis......Page 151
9.1 Euclid’s Algorithm......Page 155
9.2 Rational and Irrational Numbers as Continued Fractions......Page 156
10. How Infinite Can You Get?......Page 162
11.1 Dedekind Cuts......Page 168
11.2 Cauchy Sequences......Page 169
11.3 Completeness......Page 170
12.1 Functions......Page 174
12.2 Limits and Continuity......Page 179
12.3 Differentiation......Page 180
12.4 Integration......Page 183
13. Some Brief Remarks About the History of Analysis......Page 188
Further Reading......Page 192
Appendix 1: The Binomial Theorem......Page 198
Appendix 2: The Language of Set Theory......Page 200
Appendix 3: Proof by Mathematical Induction......Page 203
Appendix 4: The Algebra of Numbers......Page 205
Hints and Solutions to Selected Exercise......Page 207
C......Page 212
F......Page 213
L......Page 214
P......Page 215
U......Page 216
Z......Page 217
