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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)

Ramanujan's Series

The calculation of was and continues to be an important consumer of infinite series. We do not have room for the history of its computation, but we should not leave it without a few further remarks.

Modern calculations to over a trillion digits are based on far more complicated series such as the one published by S. Ramanujan in 1915:

The reader interested in details of modern calculations of is directed to “Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi” by J. M. Borwein, P. B. Borwein, and D. H. Bailey. For a popular account of the calculation of the first two billion digits of by David and Gregory Chudnovsky, see “The Mountains of Pi” by Richard Preston. Full references to these articles are in the Bibliography. More about can be found at Jonathon Borwein's Talking about Pi.

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