Ramanujan’s formula for pi
Around 1910, Ramanujan proved the following formula:
Theorem.
The following series converges and the sum equals 1π:
1π=2√29801∞∑n=0(4n)!(1103+26390n)(n!)43964n. |
Needless to say, the convergence is extremely fast. For example, if we only use the term n=0 we obtain the following approximation:
π≈98012⋅1103⋅√2=3.14159273001… |
and the error is (in absolute value) equal to 0.0000000764235… In 1985, William Gosper used this formula to calculate the first 17 million digits of π.
Another similar formula can be easily obtained from the power series of arctanx. Although the convergence is good, it is not as impressive as in Ramanujan’s formula:
π=2√3∞∑n=0(-1)n(2n+1)3n. |
Title | Ramanujan’s formula for pi |
---|---|
Canonical name | RamanujansFormulaForPi |
Date of creation | 2013-03-22 15:53:41 |
Last modified on | 2013-03-22 15:53:41 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 7 |
Author | alozano (2414) |
Entry type | Theorem |
Classification | msc 11-00 |
Classification | msc 51-00 |
Related topic | CyclometricFunctions |