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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's Series

2.4 The Harmonic Series

2.5 Taylor Series

2.6 Emerging Doubts

2.3 Calculating Pi (continued)

The Arctangent Series

The oldest and one of the most elegant series for computing is the one we saw in equation (1.4.4):

(2.3.1)

This can also be derived from the geometric series when one recalls that

(2.3.2)

We can expand the integrand using the geometric series and then integrate each term to get

(2.3.3)

This series converges very slowly, but we can use the geometric series to find the power series expansion for the arctangent of any value between 0 and 1.

(2.3.4)

The convergence is faster as we take values of x closer to 0. Around 1706, John Machin (1680–1751) used the identity

(2.3.5)
to calculate the first 100 digits of .

Many people rediscovered the series for arctan x. It is usually attributed to Leibniz but was also known to Newton and to James Gregory (1638–1675) who had independently discovered the binomial series. Almost two centuries earlier, it was known to Nilakantha (ca. 1450–1550) of Kerala in southwest India where the power series for the sine and cosine probably had been discovered even earlier by Madhava (ca. 1340–1425).

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