Ramanujan’s formula for pi

Around $1910$, Ramanujan proved the following formula:

Theorem.

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

 $\frac{1}{\pi}=\frac{2\sqrt{2}}{9801}\sum_{n=0}^{\infty}\frac{(4n)!(1103+26390n% )}{(n!)^{4}396^{4n}}.$

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:

 $\pi=2\sqrt{3}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)3^{n}}.$
Title Ramanujan’s formula for pi RamanujansFormulaForPi 2013-03-22 15:53:41 2013-03-22 15:53:41 alozano (2414) alozano (2414) 7 alozano (2414) Theorem msc 11-00 msc 51-00 CyclometricFunctions