PlanetMath: Ramanujan's formula for pi

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Ramanujan's formula for pi

(Theorem)

Around , Ramanujan proved the following formula:

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

$\displaystyle \frac{1}{\pi}=\frac{2\sqrt{2}}{9801}\sum_{n=0}^\infty \frac{(4n)!(1103+26390n)}{(n!)^4396^{4n}}.$

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

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

and the error is (in absolute value) equal to $0.0000000764235\ldots$ In

, 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:

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

"Ramanujan's formula for pi" is owned by alozano.

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See Also: cyclometric functions

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Cross-references: power series, similar, digits, calculate, absolute value, approximation, term, sum, converges, series, Ramanujan

This is version 4 of Ramanujan's formula for pi, born on 2006-05-03, modified 2007-07-01.
Object id is 7896, canonical name is RamanujansFormulaForPi.
Accessed 42079 times total.

Classification:

AMS MSC:	11-00 (Number theory :: General reference works )
	51-00 (Geometry :: General reference works )

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