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Preface

1 Crisis in Mathematics£º Fourier's Series
1£®1 Background to the Problem
1£®2 Difficulties with the Solution

2 Infinite Summations
2£®1 The Archimedean Understanding
2£®2 Geometric Series
2£®3 Calculating ¦Ð
2£®4 Logarithms and the Harmonic Series
2£®5 Taylor Series
2£®6 Emerging Doubts

3 Differentiability and Continuity
3£®1 Differentiability
3£®2 Cauchy and the Mean Value Theorems
3£®3 Continuity
3£®4 Consequences of Continuity
3£®5 Consequences of the Mean Value Theorem

4 The Convergence of Infinite Series
4£®1 The Basic Tests of Convergence
4£®2 Comparison Tests
4£®3 The Convergence of Power Series
4£®4 The Convergence of Fourier Series

5 Understanding Infinite Series
5£®1 Groupings and Rearrangements
5£®2 Cauchy and Continuity
5£®3 Differentiation and Integration
5£®4 Verifying Uniform Convergence

6 Return to Fourier Series
6£®1 Dirichlet's Theorem
6£®2 The Cauchy Integral
6£®3 The Riemann Integral
6£®4 Continuity without Differentiability

7 Epilogue
A Explorations of the Infinite
A£®1 Wallis on ¦Ð
A£®2 Bernoulli's Numbers
A£®3 Sums of Negative Powers
A£®4 The Size of n!
B Bibliography
C Hints to Selected Exercises

Index