Ramanujan’s Formula for Pi
First found by Ramanujan. It’s my favourite formula for pi. I have no idea how it works.
\[\frac{1}{\pi}
=
\frac{\sqrt{8}}{9801}
\sum_{n=0}^{\infty}\frac{(4n)!}{(n!)^4}\times\frac{26390n + 1103}{396^{4n}}\]
A Ramanujan-type formula due to the Chudnovsky brothers used to break a world record for computing the most digits of pi:
\[\frac{1}{\pi}
= \frac{1}{53360 \sqrt{640320}}
\sum_{n=0}^\infty (-1)^n
\frac{(6n)!}{n!^3(3n)!} \times
\frac{13591409 + 545140134n}{640320^{3n}}\]
For implementations, it may help to use \(640320^3 = 8\cdot 100100025\cdot 327843840\)