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Limits Limits Everywhere The Tools of Mathematical Analysis 1st Edition by David Applebaum ISBN 0191627860 9780191627866

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Limits Limits Everywhere The Tools of Mathematical Analysis 1st Edition David Applebaum Digital Instant Download

Author(s): David Applebaum
ISBN(s): 9780199640089, 0199640084
Edition: 1
File Details: PDF, 3.28 MB
Year: 2012
Language: english
SKU: EB-4348294 Category: Tag:

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Product details: 

ISBN 10: 0191627860

ISBN 13: 9780191627866

Author:  David Applebaum

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.

Table of contents: 

Part I Approaching Limits
1. A Whole Lot of Numbers
1.1 Natural Numbers
1.2 Prime Numbers
1.3 The Integers
1.4 Exercises for Chapter 1
2. Let’s Get Real
2.1 The Rational Numbers
2.2 Irrational Numbers
2.3 The Real Numbers
2.4 A First Look at Infinity
2.5 Exercises for Chapter 2
3. The Joy of Inequality
3.1 Greater or Less?
3.2 Intervals
3.3 The Modulus of a Number
3.4 Maxima and Minima
3.5 The Theorem of the Means
3.6 Getting Closer
3.7 Exercises for Chapter 3
4. Where Do You Go To, My Lovely?
4.1 Limits
4.2 Bounded Sequences
4.3 The Algebra of Limits
4.4 Fibonacci Numbers and the Golden Section
4.5 Exercises for Chapter 4
5. Bounds for Glory
5.1 Bounded Sequences Revisited
5.2 Monotone Sequences
5.3 An Old Friend Returns
5.4 Finding Square Roots
5.5 Exercises for Chapter 5
6. You Cannot be Series
6.1 What are Series?
6.2 The Sigma Notation
6.3 Convergence of Series
6.4 Nonnegative Series
6.5 The Comparison Test
6.6 Geometric Series
6.7 The Ratio Test
6.8 General Infinite Series
6.9 Conditional Convergence
6.10Regrouping and Rearrangements
6.11Real Numbers and Decimal Expansions
6.12Exercises for Chapter 6
Part II Exploring Limits
7. Wonderful Numbers – e, π and γ
7.1 The Number e
7.2 The Number π
7.3 The Number γ
8. Infinite Products
8.1 Convergence of Infinite Products
8.2 Infinite Products and Prime Numbers
8.3 Diversion – Complex Numbers and the Riemann Hypothesis
9. Continued Fractions
9.1 Euclid’s Algorithm
9.2 Rational and Irrational Numbers as Continued Fractions
10. How Infinite Can You Get?
11. Constructing the Real Numbers
11.1 Dedekind Cuts
11.2 Cauchy Sequences
11.3 Completeness
12. Where to Next in Analysis? The Calculus
12.1 Functions
12.2 Limits and Continuity
12.3 Differentiation
12.4 Integration
13. Some Brief Remarks About the History of Analysis

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