Projects & Research A Radical Approach to Real Analysis Macalester College


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


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


Examples of this are


Click here to explore the partial sums of these series.

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


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