Preface 1 Crisis in Mathematics： 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 π 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 π 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
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