How is the value of $\pi$ calculated?

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Simple numerical methods for calculating the digits of Pi

How is the value of $\pi$ calculated ?

I read, $\pi \approx 22/7$

there many formulas for approximating the value of $\pi$. They can be found [listed here.](http://mathworld.wolfram.com/PiFormulas.html) — S L, Jan 19 '13 at 04:42
My comment had links to other very closely related questions. It seems to have been deleted, but the links are still on this page over on the right. — Jonas Meyer, Jan 19 '13 at 06:30

score 7 · Answer 1 · edited Jan 19 '13 at 04:55

There are many formulas that calculate the decimals of $\pi$, here are few :

$\pi=\textstyle \cfrac{4}{1+\textstyle \frac{1^2}{2+\textstyle \frac{3^2}{2+\textstyle \frac{5^2}{2+\textstyle \frac{7^2}{2+\textstyle \frac{9^2}{2+\ddots}}}}}} =3+\textstyle \frac{1^2}{6+\textstyle \frac{3^2}{6+\textstyle \frac{5^2}{6+\textstyle \frac{7^2}{6+\textstyle \frac{9^2}{6+\ddots}}}}} =\textstyle \cfrac{4}{1+\textstyle \frac{1^2}{3+\textstyle \frac{2^2}{5+\textstyle \frac{3^2}{7+\textstyle \frac{4^2}{9+\ddots}}}}} $

$\frac{\pi}4\;=\;\sum_{k=0}^\infty\frac{(-1)^k}{2k+1}. $

This one is amongs the most rapid in term of convergence :

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

Wikipedia gives a lot of information about it.

How is the value of $\pi$ calculated?

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