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# 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

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Reference

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Theorem

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## Mathematics Subject Classification

11-00*no label found*51-00

*no label found*

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