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Ramanujan's formula for pi
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(Theorem)
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Around $1910$ , Ramanujan proved the following formula:
Theorem 1 The following series converges and the sum equals $\frac{1}{\pi}$ : $$\frac{1}{\pi}=\frac{2\sqrt{2}}{9801}\sum_{n=0}^\infty \frac{(4n)!(1103+26390n)}{(n!)^4396^{4n}}.$$
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 $\pi$
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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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"Ramanujan's formula for pi" is owned by alozano.
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Cross-references: power series, similar, digits, calculate, absolute value, approximation, term, sum, converges, series, formula, 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 50921 times total.
Classification:
| AMS MSC: | 11-00 (Number theory :: General reference works ) | | | 51-00 (Geometry :: General reference works ) |
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Pending Errata and Addenda
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