Ramanujan's formula for pi

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Ramanujan’s formula for pi

Around $1910$ , Ramanujan proved the following formula:

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}}.

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Reference

Type of Math Object:

Theorem

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Mathematics Subject Classification

11-00 General reference works (handbooks, dictionaries, bibliographies, etc.)
51-00 General reference works (handbooks, dictionaries, bibliographies, etc.)

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Ramanujan's formula for pi

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Ramanujan’s formula for pi

Theorem.

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Ramanujan's formula for pi

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Theorem.

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