# Infinite series that converges to pi

1. Feb 19, 2012

### 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.

2. Feb 19, 2012

### DaveC426913

3. Feb 19, 2012

### 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].