Infinite series that converges to pi

[keeper1]

I stumbled upon this infinite series that converges to $\pi$:

4$\sum\frac{\sqrt{n^2-i^2}}{n^2}$ for i = 1:n as n${\rightarrow∞}$

I haven't been able to find any similar series online and I'm really curious how to prove this does indeed converge to $\pi$. Any insight would be greatly appreciated.

 

[DaveC426913]

I'm not sure how similar you're looking for, but there are other series that converge on pi.

Here's a couple:
http://en.wikipedia.org/wiki/Convergent_series
and here:
http://mathworld.wolfram.com/PiFormulas.html

 

[Norwegian]

This series appear if you try to compute the area of a quarter unit circle, by approximating with n rectangles in the obvious way (one side being 1/n).

EQuivalently, your sum is a Riemann sum of f(x)=√(1-x²) in the interval [0,1].