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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

By the fourteenth century, the Scholastics in Oxford and Paris, people such as Richard Swineshead ( fl. c. 1340–1355) and Nicole Oresme (1323–1382), were using and assigning values to infinite series that arose in problems of motion. They began with series for which each pair of consecutive summands has the same ratio, such as the summation used by Archimedes,

Any series such as this for which there is a constant ratio between successive summands is called a geometric series. For many values of x, the infinite geometric series can be summed using the identity

(2.2.1)

Examples of this are

and

One has to be careful with equation (2.2.1). If we set x = 2, we get a very strange equality:

(2.2.2)

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