Finding an accurate approximation to has been one of the most noteworthy challenges in the history of mathematics. Srinivasa A. Ramanujan (1887–1920), a mathematical thinker of phenomenal abilities, discovered a mysterious infinite series for estimating the value of [1]:

.

The series is known to be a specialization of a modular equation of order 58 [2].

This Demonstration gives numerical estimates for using the reciprocal of the series up to , which gives a correct approximation to 38 decimal places.

[1] S. Ramanujan, "Modular Equations and Approximations to ," The Quarterly Journal of Mathematics, 45, 1914 pp. 350–372.

[2] J. M. Borwein, P. B. Borwein and D. H. Bailey, "Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi," The American Mathematical Monthly, 96(3), 1989 pp. 201–219. doi:10.2307/2325206.