Why the $\lim_{n\to\infty} (\frac{n}{n^2+1}+\frac{n}{n^2+4}+.....+\frac{n}{n^2+n^2})= \frac{\pi}{4}$? Why the $\lim_{n\to\infty} (\frac{n}{n^2+1}+\frac{n}{n^2+4}+.....+\frac{n}{n^2+n^2})= \frac{\pi}{4}$?
I read somewhere that it is related to $f(x)=\frac{1}{1+x^2}$ but dont know why...

Use the def of Riemann integration. Write 
$$ \lim_{n \to \infty} \sum_{i=1}^n \frac{ n}{n^2+ i^2} = \lim_{n \to \infty} \sum_{1}^n \frac{1/n}{1+(i/n)^2} = \int\limits_0^1 \frac{1}{1+x^2} dx = \frac{\pi}{4} $$
