Infinite Series

Infinite Series

by ABC-CLIO

Hardcover(Revised ed.)

$86.00
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Overview

This text for advanced undergraduate and graduate students presents a rigorous approach that also emphasizes applications. Encompassing more than the usual amount of material on the problems of computation with series, the treatment offers many applications, including those related to the theory of special functions. Numerous problems appear throughout the book.
The first chapter introduces the elementary theory of infinite series, followed by a relatively complete exposition of the basic properties of Taylor series and Fourier series. Additional subjects include series of functions and the applications of uniform convergence; double series, changes in the order of summation, and summability; power series and real analytic functions; and additional topics in Fourier series. The text concludes with an appendix containing material on set and sequence operations and continuous functions.



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Product Details

ISBN-13: 9780837198972
Publisher: ABC-CLIO, Incorporated
Publication date: 02/10/1978
Series: Athena Series, Selected Topics in Mathematics Series
Edition description: Revised ed.
Pages: 173
Product dimensions: 6.00(w) x 9.00(h) x 0.50(d)

About the Author

Isidore Isaac Hirschman, Jr. (1922–90) was a Professor of Mathematics at Washington University in St. Louis. Dover also publishes a book he co-authored with David V. Widder, The Convolution Transform.

Table of Contents

Chapter 1 Tests for Convergence and Divergence 1

1 Sequences 1

1* Limits 3

2 Convergence 6

3 Some Preliminary Results 9

4 Integral Test 12

4* Error Estimates 16

5 Comparison Test 18

5* Error Estimates Continued 23

6 Relative Magnitudes 24

7 Absolute Convergence 27

8 Alternating Series 28

9 Power Series 31

10 Additional Exercises 34

Chapter 2 Taylor Series 36

1 Introduction 36

2 Elementary Transformations 42

3 The Remainder Formula 47

4 The Remainder Formula (Continued) 52

Chapter 3 Fourier Series 56

1 Classes of Functions 56

2 Fourier Series 59

3 Bessels's Inequality 64

4 Dirichlet's Formula 67

5 A Convergence Theorem 70

6 Sine and Cosine Series 74

7 Complex Fourier Series 77

Chapter 4 Uniform Convergence 82

1 Sequences of Functions 82

2 Uniform Convergence of Continuous Functions 87

3 Miscellaneous Results 95

4 Summation by Parts 97

Chapter 5 Rearrangements, Double Series, Summability 103

1 Generalized Partial Sums 103

2 Double Series 107

3 Products 111

4 Fubini's Theorem 119

5 Summability 123

6 Regularity 128

Chapter 6 Power Series and Real Analytic Functions 131

1 Radius of Convergence 131

2 Operations on Power Series 133

3 End Point Behavior 140

4 Real Analytic Functions 143

Chapter 7 Additional Topics in Fourier Series 149

1 (C, 1) Summability 149

2 Uniform Continuity 153

3 Parseval's Equality 155

4 Convolution 158

Appendix 163

1 Set and Sequence Operations 163

2 Continuous Functions 166

Index 171

Index of Symbols 173

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