Ramanujan’s formula for pi

Around 1910, Ramanujan proved the following formula:


The following series converges and the sum equals 1π:


Needless to say, the convergence is extremely fast. For example, if we only use the term n=0 we obtain the following approximation:


and the error is (in absolute valueMathworldPlanetmathPlanetmathPlanetmath) 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 seriesMathworldPlanetmath of arctanx. Although the convergence is good, it is not as impressive as in Ramanujan’s formula:

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