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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)

Problems with Infinite Series

If the trouble had only lain in the definition of the derivative and integral, then it would not have received the attention that it did. Infinite series were also causing misgivings. Euler worked with divergent series and, as we saw in section 2.2, determined the value from the genesis of the series. He would assign the value to the divergent series
because it arises from the Taylor series for .

There is a difficulty with this point of view that was exposed by Jean Bernoulli's son Daniel (1700–1782) in 1772: different machinery can give rise to the same series with different values. The alternating series of 1's and –1's can arise when x is set equal to 1 in

or

or

The first gives a value of 1/2, the second 1/3, and the last 2/3. We can vary these exponents to get any rational number between 0 and 1. The same divergent series can have different values depending upon the context in which it arises. Many mathematicians found this to be a highly unsatisfactory state of affairs. On the other hand, to simply discard all divergent series is to lose those, like the error term in Stirling's formula, that are truly useful.

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