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Chapter 1: Crises in Mathematics: Fourier's Series

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 (continued)

Wallis's Product

John Wallis (1616–1703) considered the integral

When p = q = 1/2, this is the area in the first quadrant located below the graph of the upper half of the unit circle centered at the origin, It equals /4.

Wallis knew the binomial theorem for integer exponents, and he knew how to integrate a rational power of x. Relying on what happens at integer values of q, he was able to extrapolate to other values. From the patterns he observed, he discovered a remarkable formula for :

(2.3.6)

Again, this is a terrible way to calculate , but it is a beautiful formula with important uses, including estimation of the size of n!.

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