Ramanujan Series

A remarkable series discovered by the Indian mathematician Srinivasa Ramanujan. It was one of the most accurate methods of determining pi to a large amount of decimal places until the advent of the Chudnovsky Series. It is still an amazing discovery, and a very elegant representation of pi. This series adds roughly eight digits per term. Gosper in 1985 computed 17 million terms of the continued fraction for pi using this.

1/pi = 2sqrt(2) / 9801 * (sigma; n=0 to infinity) [( (4n)! / n!^4) * (1103 + 26390n) / (4 * 99)^4n]

I suppose it loses some of it's elegance when represented in plain ascii on html, but hey...the symmetry is beautiful, and the simplicity is beguiling.

Srinivasa Ramanujan, a brilliant Indian mathematician who died in the early 20th century, devised this infinite series which would compute the reciprocal value of pi in linear time, adding eight digits to the value with every iteration. This was considerably faster than the usual arctangent method, but not as fast as the Brent-Salamin algorithm discovered in 1976.

This formula was first discovered in 1910, but was not widely known or used until 1985 when it was used in a supercomputer (along with some FFT code to speed up the multiplication) to successfully calculate pi to 17 million digits.

Ramanujan devised other infinite series that would do the same job, but this one is the fastest and best-known. Although the accuracy of this formula can be and has been proven mathematically, it's believed that Ramanujan himself devised this formula by brute force and observing the results.

The formula itself is:

        __  oo
1    2 V 2 \   (4k)!(1103+26390k)
-- = ----- /   ------------------
pi   9801  k=0    (k!)4 3964k