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Chapter 1: Crises in Mathematics: Fourier's Series

Chapter 2: Infinite Summations

2.1 Avoiding Infinite Summations

2.2 Geometric Series

> A Question of Definition

Cauchy's Approach

2.3 Calculating Pi

2.4 The Harmonic Series

2.5 Taylor Series

2.6 Emerging Doubts

Exercises

2.2 Geometric Series (continued)

A Question of Definition

We need to decide what we mean by an infinite summation. We could define to mean in which case equation (2.2.2) is correct. We would be in good company. Leonhard Euler accepted this definition. It yields many other interesting results, for example:

(2.2.4)
In the exhaustive and fascinating account, Convolutions in French Mathematics, 1800–1840, Ivor Grattan-Guinness writes, “Some modern appraisals of the cavalier style of 18th-century mathematicians in handling infinite series convey the impression that these poor men set their brains aside when confronted by them.” They did not. Certainly Euler had not set his brain aside. He rather viewed infinite series in a larger context, a context that he makes clear in his article “On divergent series” published in 1760. Euler illustrates his understanding with the series which he asserts to be equal to 1/2, obtained by setting x = –1 in equation (2.2.1).

“Notable enough, however, are the controversies over the series 1 – 1 + 1 – 1 + 1 – etc. whose sum was given by Leibniz as 1/2, although others disagree ... Understanding of this question is to be sought in the word 'sum'; this idea, if thus conceived—namely, the sum of a series is said to be that quantity to which it is brought closer as more terms of the series are taken—has relevance only for the convergent series, and we should in general give up this idea of sum for divergent series. On the other hand, as series in analysis arise from the expansion of fractions or irrational quantities or even of transcendentals, it will in turn be permissible in calculation to substitute in place of such series that quantity out of whose development it is produced.”

Here is the point we have been making: for any infinite summation we need to stretch our definition of sum. Euler merely asks that in the case of a series that does not converge, we allow a value determined by the genesis of the series.

As we shall see in section 2.6, Euler's approach raises more problems than it settles. Eventually, mathematicians wouyld be forced to allow divergent series to have values. such values are too useful to abandon completely. But using these values must be done with great delicacy. The scope of this book will only allow brief glimpses of how this can be done safely. Archimedes' solution is the easiest and most reliable way of assigning values to infinite series.

When an infinite series has a target value in the sense of Archimedes' understanding, we say that our series converges. For our purposes, it will be safest not to assign a value to an infinite series unless it converges.

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