Formulas

The most beautiful formulas:

Ramanujan's inverse pi formula:

$\pi ^{-1}=\displaystyle {\dfrac {\sqrt {8}}{99^{2}}}\displaystyle {\sum _{k\geq 0}{\dfrac {(4k)!}{(4^{k}k!)^{4}}}}\displaystyle {\dfrac {1103+26390k}{99^{4k}}}$

Gaussian integral:

$\int _{-\infty }^{\infty }e^{-x^{2}}dx={\sqrt {2\pi }}$

Ramanujan sum:

$\sum _{k=1}^{\infty }k=-{\frac {1}{12}}$

Euler formula [1]:

$e^{i\pi }+1=0$

Euler's sum of inverse squares:

$\sum _{n=1}^{\infty }{\dfrac {1}{n^{2}}}={\dfrac {\pi ^{2}}{6}}$

Archimedes' Recurrence Formula:

$a_{2n}={\frac {2a_{n}b_{n}}{a_{n}+b_{n}}}$

$b_{2n}={\sqrt {a_{2n}b_{n}}}$