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Chapter 1: Crises in Mathematics: Fourier's Series

Chapter 2: Infinite Summations

2.1 Avoiding Infinite Series

2.2 The Geometric Series

2.3 Calculating Pi

2.4 The Harmonic Series

2.5 Taylor Series

2.6 Emerging Doubts

Problem with Series

Vibrating String

>Cauchy's Counter-example

2.6 Emerging Doubts (continued)

Cauchy's Counterexample

The death knell for Lagrange's definition of the derivative was sounded by Cauchy in 1821. He exhibited a counterexample to Lagrange's assertion that distinct functions have distinct power series:

All of the derivatives of f(x) at x=0 are equal to 0. At x=0, this function has the same power series expansion as the constant function 0. The determination of the derivatives of f(x) at x=0 will be demonstrated in section 3.3.

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