Real Analysis via Sequences and Series

Başlık:

Yazar:

Little, Charles H.C. author.

ISBN:

9781493926510

Ek Yazar:

Little, Charles H.C. author.

Fiziksel Tanım:

XI, 476 p. 27 illus. online resource.

Series:

Undergraduate Texts in Mathematics,

Contents:

Preface -- 1. Introduction -- 2. Sequences -- 3. Series -- 4. Limits of Functions -- 5. Continuity -- 6. Differentiability -- 7. The Riemann Integral -- 8. Taylor Polynomials and Taylor Series -- 9. The Fixed Point Problem -- 10. Sequences of Functions -- Bibliography -- Index.

Abstract:

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.

Konu Terimleri:

Mathematics.

Functions of real variables.

Sequences (Mathematics).

Real Functions.

Sequences, Series, Summability.

Added Author:

Teo, Kee L.

van Brunt, Bruce.

Added Corporate Author:

SpringerLink (Online service)

Electronic Access:

http://dx.doi.org/10.1007/978-1-4939-2651-0

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Library	Materyal Türü	Barkod	Yer Numarası	Durum
Searching... Pamukkale Merkez Kütüphanesi	E-Kitap	73984-1001	QA331.5	Searching... Unknown

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Özet

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.

Author Notes

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

Reviews (1)

Choice Review

Courses in real analysis are taken by most math majors and by majors in some disciplines in which background in a rigorous approach to math is useful. One key goal of such a course is to help students develop mathematical maturity. Intended as a textbook in a third-year course in real analysis, this book differentiates itself from other textbooks by flipping the order of the material. Sequences and series are part of the syllabus for most course offerings, but normally as self-contained discussions near the end of the course. The authors (all, mathematics, Massey Univ., New Zealand) introduce sequences and series at the beginning and build the fundamental concepts of analysis from them. This method is nontraditional, but it achieves the same goal of introducing students to mathematical rigor and basic concepts and results in real analysis. Though the book is primarily a textbook, some students might wish to see the material from a different perspective. That being the case, this title can also be used as a resource outside the classroom. Summing Up: Recommended. Upper-division undergraduates. --Donald Z. Spicer, University System of Maryland