Ramanujan's Strange Formula for Pi

terms in summation

0

1

2

3

4

1

π

≈

12389404344116838964769400621280991

1131379202490552979877435552947122965839872

2

+

2206

2

9801

approximate value of π:

3.1415926535897932384626433832795028841976638181330

exact expansion to 50 digits:

3.1415926535897932384626433832795028841971693993751

Finding an accurate approximation to

π

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

π

[1]:

1

π

=

8

9801

∞

∑

n=0

(4n)!(26390n+1103)

4

(n!)

4n

396

.

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

This Demonstration gives numerical estimates for

π

using the reciprocal of the series up to

n=4

, which gives a correct approximation to 38 decimal places.