How to find the correct value of pi?

Question

Pi is defined as the ratio of $\frac{c}{r}$. Many ancient scintist try to find the value of pi. Some of the values are $\frac{22}{7}$(good hold upto 10 decimal point), $\frac{355}{113}$ (good hold upto 63 Decimal point). modern computer determine upto million decimal point.

my qustion is that how ( algorithm or formula ) modern computer find the value of pi ?

Nop, i dont't want any guss. I need exect way of algo. That modern computer follow to find exect value of pi. — Prashant, Apr 22 '16 at 04:40
This Wikipedia article has a lot of information, and additional links. — André Nicolas, Apr 22 '16 at 04:43
If you click on the "(pi)" tag, and then select "votes" to get the highest voted questions, you will find several questions that answer this. More examples: http://math.stackexchange.com/questions/14113/series-that-converge-to-pi-quickly, http://math.stackexchange.com/questions/1189820/calculating-pi-manually, ... — Hans Lundmark, Apr 22 '16 at 04:47
Also, $\pi$ is not "defined as the ratio $\frac{c}{r}$." It is $\frac{c}{d}$. — JRN, Apr 22 '16 at 04:51
Why do you say $22/7$ is a "good hold upto 10 decimal point"? $\pi=3.141592...$ and $22/7=3.142857...$ and they differ at the third decimal place. — JRN, Apr 22 '16 at 04:57

Pencil · Accepted Answer · 2016-04-22T15:59:56.693

There are many formulas after the development of calculus, mostly in the form of infinite series. For a simple example, it is a fact that $$\pi=4\left(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots\right).$$ (This can be seen through the Taylor series of arctan x). This is a simple example, but it converges really slowly; you need about 100 terms to get the 3.14 value. There are more sophisticated formulas, but I hope this gave you the idea.

How to find the correct value of pi?

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