David Applebaum
Oxford University Press | 2012 | 217 páginas | RAR - PDF | 1,09 Mb
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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 ?, 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.
Contents
PART I APPROACHING LIMITS
1. A Whole Lot of Numbers 2
1.1 Natural Numbers 2
1.2 Prime Numbers 3
1.3 The Integers 9
1.4 Exercises for Chapter 1 12
2. Let’s Get Real 14
2.1 The Rational Numbers 14
2.2 Irrational Numbers 17
2.3 The Real Numbers 23
2.4 A First Look at Infinity 24
2.5 Exercises for Chapter 2 26
3. The Joy of Inequality 28
3.1 Greater or Less? 28
3.2 Intervals 33
3.3 The Modulus of a Number 34
3.4 Maxima and Minima 38
3.5 The Theorem of the Means 38
3.6 Getting Closer 41
3.7 Exercises for Chapter 3 42
4. Where Do You Go To, My Lovely? 45
4.1 Limits 45
4.2 Bounded Sequences 54
4.3 The Algebra of Limits 56
4.4 Fibonacci Numbers and the Golden Section 59
4.5 Exercises for Chapter 4 62
5. Bounds for Glory 65
5.1 Bounded Sequences Revisited 65
5.2 Monotone Sequences 69
5.3 An Old Friend Returns
5.4 Finding Square Roots 73
5.5 Exercises for Chapter 5 75
6. You Cannot be Series 78
6.1 What are Series? 78
6.2 The Sigma Notation 79
6.3 Convergence of Series 82
6.4 Nonnegative Series 84
6.5 The Comparison Test 88
6.6 Geometric Series 92
6.7 The Ratio Test 95
6.8 General Infinite Series 98
6.9 Conditional Convergence 99
6.10 Regrouping and Rearrangements 102
6.11 Real Numbers and Decimal Expansions 104
6.12 Exercises for Chapter 6 105
PART II EXPLORING LIMITS
7. Wonderful Numbers – e,π and γ 110
7.1 The Number e 110
7.2 The Number π 118
7.3 The Number γ 123
8. Infinite Products 126
8.1 Convergence of Infinite Products 126
8.2 Infinite Products and Prime Numbers 130
8.3 Diversion – Complex Numbers and the Riemann Hypothesis 134
9. Continued Fractions 138
9.1 Euclid’s Algorithm 138
9.2 Rational and Irrational Numbers as Continued Fractions 139
10. How Infinite Can You Get? 145
11. Constructing the Real Numbers 151
11.1 Dedekind Cuts 151
11.2 Cauchy Sequences 152
11.3 Completeness 153
12. Where to Next in Analysis? The Calculus 157
12.1 Functions 157
12.2 Limits and Continuity
12.3 Differentiation 163
12.4 Integration 166
13. Some Brief Remarks About the History of Analysis 171
Further Reading 175
Appendices 181
Appendix 1: The Binomial Theorem 181
Appendix 2: The Language of Set Theory 183
Appendix 3: Proof by Mathematical Induction 186
Appendix 4: The Algebra of Numbers 188
Hints and Solutions to Selected Exercise 190
Index 195
