# Math Help - proving limit of sum of series

1. ## proving limit of sum of series

Show that
$\lim_{n\to\infty}\left( \sum_{k=1}^{n} \frac{n}{n^2+k^2}\right ) = \frac{\pi}{4}$

Can we use integral test to prove this? How do we go about doing this?

2. ## Re: proving limit of sum of series

Hello alphabeta89! Do you that $\int_{0}^{1} f(x) dx=\lim_{n \to \infty} \sum_{k=1}^{n}\frac{1}{n}{f \Big(\dfrac{k}{n} \Big)$

$\lim_{n \to \infty}\sum_{k=1}^{n}\frac{n}{n^2+k^2}=\lim_{n \to \infty}\sum_{k=1}^{n}\frac{1}{n}\frac{1}{1+\frac{k ^2}{n^2}}$

$=\int_{0}^{1}\frac{1}{1+x^2}dx=\Big( \arctan x\Big) \Big|_{0}^{1}$

$=\frac{\pi}{4}$