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
