Limits, Limits Everywhere

The Tools of Mathematical Analysis

David Applebaum (University of Sheffield)

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English

Oxford University Press
27 March 2012

Mathematics & Sciences; Calculus & mathematical analysis; Real analysis, real variables

"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
*p, 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."

By: David Applebaum (University of Sheffield)
Imprint: Oxford University Press
Country of Publication: United Kingdom
Dimensions: Height: 234mm, Width: 153mm, Spine: 12mm
Weight: 347g
ISBN: 9780199640089
ISBN 10: 0199640084
Pages: 256
Publication Date: 27 March 2012
Audience: College/higher education , General/trade , Primary , ELT Advanced
Format: Paperback
Publisher's Status: Active

Introduction I Approaching Limits 1: A Whole Lot of Numbers 2: Let's Get Real 3: The Joy of Inequality 4: Where Do You Go To, My Lovely 5: Bounds for Glory 6: You Cannot be Series II Exploring Limits 7: Wonderful Numbers 8: Infinite Products 9: Continued Fractions 10: How Infinite Can You Get? 11: Constructing the Real Numbers 12: Where to Next in Analysis? The Calculus 13: Some Brief Remarks About the History of Analysis Further Reading Apendices 1: The Binomial Theorem 2: The Language of Set Theory 3: Proof by Mathematical Induction 4: The Algebra of Numbers Hints and Selected Solutions

Reviews for Limits, Limits Everywhere: The Tools of Mathematical Analysis

The book is devoted to the discussion of one of the most difficult concepts of mathematical analysis, the concept of limits. The presentation is instructive and informal. It allows the author to go much deeper than is usually possible in a standard course of calculus. Moreover, each portion of the material is supplied by an explanation why and what for it is necessary to study (and to teach) the corresponding part of calculus ... the book can be recommended for interested students as well as for teachers in mathematics. * Zentralblatt MATH * This is an excellent book which should appeal to teachers and pre-University or undergraduate students looking for a hands-on introduction to mathematical analysis. * Mario Cortina Borja, Significance * Recommended in the Times Higher Education's Textbook Guide 2012. * Noel-Ann Bradshaw, Times Higher Education * Written in a style that is easy to read and follow, the author gives clear and succinct explanations and meets his desire for this to be between a textbook and a popular book on mathematics. * John Sykes, Mathematics in Schools * The author is able to mix both styles relating informal language to mathematical language and giving proofs that are deep but easy to read and follow. * Luis Sanchez-Gonzalez, the European Mathematical Society * This book does not offer an easy ride but its informal and enthusiastic literary style hold ones attention. Perhaps mindful of the content of much current popular mathematical exposition, the author draws many illustrations from number theory. * Geoffrey Burton, LMS Newsletter *