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© 2015

Real Analysis via Sequences and Series

Textbook

Part of the Undergraduate Texts in Mathematics book series (UTM)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Charles H. C. Little, Kee L. Teo, Bruce van Brunt
    Pages 1-32
  3. Charles H. C. Little, Kee L. Teo, Bruce van Brunt
    Pages 33-108
  4. Charles H. C. Little, Kee L. Teo, Bruce van Brunt
    Pages 109-189
  5. Charles H. C. Little, Kee L. Teo, Bruce van Brunt
    Pages 191-214
  6. Charles H. C. Little, Kee L. Teo, Bruce van Brunt
    Pages 215-241
  7. Charles H. C. Little, Kee L. Teo, Bruce van Brunt
    Pages 243-332
  8. Charles H. C. Little, Kee L. Teo, Bruce van Brunt
    Pages 333-398
  9. Charles H. C. Little, Kee L. Teo, Bruce van Brunt
    Pages 399-421
  10. Charles H. C. Little, Kee L. Teo, Bruce van Brunt
    Pages 423-436
  11. Charles H. C. Little, Kee L. Teo, Bruce van Brunt
    Pages 437-469
  12. Back Matter
    Pages 471-476

About this book

Introduction

This text gives a rigorous treatment of the foundations of calculus. In contrast to more traditional approaches, infinite sequences and series are placed at the forefront. The approach taken has not only the merit of simplicity, but students are well placed to understand and appreciate more sophisticated concepts in advanced mathematics. The authors mitigate potential difficulties in mastering the material by motivating  definitions, results, and proofs. Simple examples  are provided to  illustrate new material and exercises are included at the end of most sections. Noteworthy topics include: an extensive discussion of convergence tests for infinite series, Wallis’s formula and Stirling’s formula, proofs of the irrationality of π and e, and a treatment of Newton’s method as a special instance of finding fixed points of iterated functions.

Keywords

Airy function Airy's equation Baire's theorem Bolzano-Weierstrass theorem Cartesian product Cauchy condensation test Dirichlet's test Kummer-Jensen test Riemann integral Sequences infinite series integral test limits of functions real analysis text adoption sequence convergence

Authors and affiliations

  1. 1.Institute of Fundamental SciencesMassey UniversityPalmerston NorthNew Zealand
  2. 2.Institute of Fundamental SciencesMassey UniversityPalmerston NorthNew Zealand
  3. 3.Institute of Fundamental SciencesMassey UniversityPalmerston NorthNew Zealand

About the authors

Charles Little, Teo Kee and Bruce van Brunt are professors of Mathematics at Massey University in New Zealand.

Bibliographic information

  • Book Title Real Analysis via Sequences and Series
  • Authors Charles H.C. Little
    Kee L. Teo
    Bruce van Brunt
  • Series Title Undergraduate Texts in Mathematics
  • Series Abbreviated Title Undergraduate Texts Mathematics
  • DOI https://doi.org/10.1007/978-1-4939-2651-0
  • Copyright Information Springer Science+Business Media New York 2015
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-1-4939-2650-3
  • Softcover ISBN 978-1-4939-4181-0
  • eBook ISBN 978-1-4939-2651-0
  • Series ISSN 0172-6056
  • Series E-ISSN 2197-5604
  • Edition Number 1
  • Number of Pages XI, 476
  • Number of Illustrations 27 b/w illustrations, 0 illustrations in colour
  • Topics Real Functions
    Sequences, Series, Summability
  • Buy this book on publisher's site

Reviews

“The list of main topics covered is quite standard: sequences, series, limits, continuity, differentiation, Riemann integration, uniform convergence … . This is a well-written textbook with an abundance of worked examples and exercises that is intended for a first course in analysis with modest ambitions.” (Brian S. Thomson, Mathematical Reviews, March, 2016)

“The authors … introduce sequences and series at the beginning and build the fundamental concepts of analysis from them. … it achieves the same goal of introducing students to mathematical rigor and basic concepts and results in real analysis. … Summing Up: Recommended. Upper-division undergraduates.” (D. Z. Spicer, Choice, Vol. 53 (5), January, 2016)

“This textbook is based on the central idea that concepts such as continuity, differentiation and integration are approached via the concepts of sequences and series. … Most of the sections are followed by exercises. The textbook is recommended for a first course in mathematical analysis.” (Sorin Gheorghe Gal, zbMATH, Vol. 1325.26002, 2016)