I remember that pi's value is a infinite series.
But how to prove that?

Well,
the series for arctan is
arctan(x) = x - x³ /3 + x⁵ /5 - x⁷ /7 + x⁹ /9 - ......

putting x=1 we get,
arctan(1) = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ....

but $\arctan (1)=\pi /4$

hence,

$\pi /4=1-1/3+1/5-1/7+1/9-\dots$

hence,

$\pi =4-4/3+4/5-4/7+4/9-\dots$

I hope this is what you asked for!!!

love arun