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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

The Arctangent Series

Wallis's Product

Newton's Binomial Series

Ramanujan, Sines, and Cosines

2.4 The Harmonic Series

2.5 Taylor Series

2.6 Emerging Doubts

2.3 Calculating Pi

Fortunately, the fear of the infinite was overcome. At first hesitantly and then with increasing confidence, mathematicians plunged into the infinite and resurfaced with treasures that Archimedes could never have imagined. The true power of calculus lies in its coupling with infinite processes. Mathematics as we know it and as it has come to shape modern science could never have come into being without a reckless disregard for the dangers of the infinite.

As we saw in the last two sections, the dangers are real. The genius of the early explorers of calculus lay in their ability to sense when they could treat an infinite summation according to the rules of the finite and when they could not. Sixth sense is a poor foundation for mathematics. By the time Fourier proposed his trigonometric series, it was recognized that a better understanding of what was happening—what was legitimate and what would lead to error—was needed. The solution that was ultimately accepted is the Archimedean understanding, but it would be a mistake to jump directly from Archimedes to the modern era for it would miss the point of that revolution in mathematics that occured in the late seventeenth century and that was so powerful precisely because it dared to treat the infinite as if it obeyed the same laws as the finite.

The time will come when we need careful definitions, when we start to find the problems and need to learn how to avoid them. These definitions will have no meaning unless we first appreciate the usefulness of being able to play with infinite summations as if they really are summations. We begin by seeing what they can do for us.

Much of the initial impetus for using the infinite came from the search for better approximations to , the ratio of the circumference of a circle to its diameter. In this section, we will describe several different infinite series, as well as an infinite product, that can be used to approximate .

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