Ramanujan's formula for piRamanujansFormulaForPi2006-05-03 09:05:102007-07-01 18:31:14Theorempi% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Cl}{\operatorname{Cl}}Around $1910$, Ramanujan proved the following formula:
\begin{thm}
The following series converges and the sum equals $\frac{1}{\pi}$:
$$\frac{1}{\pi}=\frac{2\sqrt{2}}{9801}\sum_{n=0}^\infty \frac{(4n)!(1103+26390n)}{(n!)^4396^{4n}}.$$
\end{thm}
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:
$$\pi=2\sqrt{3}\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)3^n}.$$