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.

References

[1] S. Ramanujan, "Modular Equations and Approximations to
π
," 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.

External Links

Pi (Wolfram MathWorld)
Pi Formulas (Wolfram MathWorld)
Pi Approximations (Wolfram MathWorld)
Pi Digits (Wolfram MathWorld)

Permanent Citation

Allan Zea, Dr. Jean Carlos Liendo
​
​"Ramanujan's Strange Formula for Pi"​
​http://demonstrations.wolfram.com/RamanujansStrangeFormulaForPi/​
​Wolfram Demonstrations Project​
​Published: February 17, 2017