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Simple numerical methods for calculating the digits of Pi

How is the value of $\pi$ calculated ?

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


marked as duplicate by Jonas Meyer, Sasha, user17762, Erick Wong, jspecter Jan 19 '13 at 5:14

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  • $\begingroup$ there many formulas for approximating the value of $\pi$. They can be found listed here. $\endgroup$ – Santosh Linkha Jan 19 '13 at 4:42
  • $\begingroup$ 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. $\endgroup$ – Jonas Meyer Jan 19 '13 at 6:30

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.


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