# Ramanujan’s formula for pi

Around $1910$, Ramanujan proved the following formula:

###### Theorem.

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

$$\frac{1}{\pi}=\frac{2\sqrt{2}}{9801}\sum _{n=0}^{\mathrm{\infty}}\frac{(4n)!(1103+26390n)}{{(n!)}^{4}{396}^{4n}}.$$ |

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

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

and the error is (in absolute value^{}) equal to $0.0000000764235\mathrm{\dots}$ 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 $\mathrm{arctan}x$. Although the convergence is good, it is not as impressive as in Ramanujan’s formula:

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

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 |