Limits, Limits Everywhere: The Tools of Mathematical Analysis

Paperback | April 15, 2012

byDavid Applebaum

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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, inparticular, 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 aset 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 p, continued fractions, an introductionto 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 inductionare 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 Riemannhypothesis, 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.

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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, inparticular, will focus on numbers, sequences,...

David Applebaum obtained his PhD at the University of Nottingham in 1984. After postdoctoral appointments in Rome and Nottingham, he became a lecturer in mathematics at Nottingham Trent University (then Trent Polytechnic) in 1987 and was promoted to reader in 1994 and to a chair in 1998. He was Head of Department 1998-2001. He left Not...

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Format:PaperbackDimensions:256 pages, 9.21 × 6.02 × 0.68 inPublished:April 15, 2012Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0199640084

ISBN - 13:9780199640089

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Table of Contents

IntroductionI Approaching Limits1. A Whole Lot of Numbers2. Let's Get Real3. The Joy of Inequality4. Where Do You Go To, My Lovely5. Bounds for Glory6. You Cannot be SeriesII Exploring Limits7. Wonderful Numbers8. Infinite Products9. Continued Fractions10. How Infinite Can You Get?11. Constructing the Real Numbers12. Where to Next in Analysis? The Calculus13. Some Brief Remarks About the History of AnalysisFurther ReadingApendices1. The Binomial Theorem2. The Language of Set Theory3. Proof by Mathematical Induction4. The Algebra of NumbersHints and Selected Solutions