Limits, Limits Everywhere
The Tools of Mathematical Analysis
1st Edition
Published on 1. March 2012
217 pages
978-0-19-162786-6 (ISBN)
Description
An account of elementary real analysis positioned between a popular mathematics book and a first year college or university text. This book doesn't assume knowledge of calculus and, instead, the emphasis is on the application of analysis to number theory.
More details
Other editions
Additional editions
Book
03/2012
1st Edition
Oxford University Press
€45.87
Shipment within 15-20 days
Content
- Cover
- 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
- 6.10 Regrouping and Rearrangements
- 6.11 Real Numbers and Decimal Expansions
- 6.12 Exercises for Chapter 6
- Part II Exploring Limits
- 7 Wonderful Numbers - e, p and ?
- 7.1 The Number e
- 7.2 The Number p
- 7.3 The Number ?
- 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
- Further Reading
- Appendices
- Appendix 1: The Binomial Theorem
- Appendix 2: The Language of Set Theory
- Appendix 3: Proof by Mathematical Induction
- Appendix 4: The Algebra of Numbers
- Hints and Solutions to Selected Exercises
- Index
