Ramanujan’s formula for pi

Around $1910$, Ramanujan proved the following formula:

Theorem.

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

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

Needless to say, the convergence is extremely fast. For example, if we only use the term $n=0$ we obtain the following approximation:

$$\pi \approx \frac{9801}{2\cdot 1103\cdot \sqrt{2}}=3.14159273001\mathrm{\dots }$$

and the error is (in absolute value) equal to $0.0000000764235\mathrm{\dots }$ 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:

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