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Chapter 1: Crises in Mathematics: Fourier's Series

2.1 Avoiding Infinite Series

2.2 The Geometric Series

2.3 Calculating Pi

2.4 The Harmonic Series

2.5 Taylor Series

>Taylor's Formula

d'Alembert

Lagrange's Remainder

2.6 Emerging Doubts

2.5 Taylor Series (continued)

Taylor's Formula

The machine described by Taylor expresses the coefficients of the power series in terms of the derivatives at a particular point. If

and assuming that we can differentiate a power series just like an ordinary polynomial, then

This implies that and therefore

In general, we can express f(x) as a sum of powers of (x-a):

(2.5.1)

All of the power series we have encountered so far are special cases of equation (2.5.1). For example, if f(x) = ln x, we observe that

We take the function f(x) = ln x, replace x by 1+x, set a = 1, and make this substitution into equation (2.5.1). This yields the power series

We have not proven anything yet. We do not know that we can differentiate a power series as if it were a polynomial. We do not even know whether or not the function the function with which we started has a power series. We assumed our function had a power series, we assumed we could differentiate it as if it were a polynomial, and that led us to conclude what the coefficients would have to be.

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