## You are here

HomeRamanujan's formula for pi

## Primary tabs

# Ramanujan’s formula for pi

###### Theorem.

The following series converges and the sum equals $\frac{1}{\pi}$:

$\frac{1}{\pi}=\frac{2\sqrt{2}}{9801}\sum_{{n=0}}^{\infty}\frac{(4n)!(1103+2639% 0n)}{(n!)^{4}396^{{4n}}}.$ |

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

$\pi\approx\frac{9801}{2\cdot 1103\cdot\sqrt{2}}=3.14159273001\ldots$ |

and the error is (in absolute value) equal to $0.0000000764235\ldots$ In $1985$, William Gosper used this formula to calculate the first 17 million digits of $\pi$.

Another similar formula can be easily obtained from the power series of $\arctan x$. Although the convergence is good, it is not as impressive as in Ramanujan’s formula:

$\pi=2\sqrt{3}\sum_{{n=0}}^{\infty}\frac{(-1)^{n}}{(2n+1)3^{n}}.$ |

Related:

CyclometricFunctions

Major Section:

Reference

Type of Math Object:

Theorem

Parent:

## Mathematics Subject Classification

11-00*no label found*51-00

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff