The limit of a sum $\sum_{k=1}^n \frac{n}{n^2+k^2}$ as $n\rightarrow\infty$ [closed]
How to compute this limit:
$$\lim_{n\to\infty}\left(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\cdots+\frac{n}{n^2+n^2}\right)$$
Please give me some hint.
$$\begin{align}\lim_{n\to\infty}\sum_{1\le r\le n}\frac{n}{n^2+r^2} &=\lim_{n\to\infty}\frac1n\sum_{1\le r\le n}\frac1{1+\left(\frac rn\right)^2}\\ &=\int_0^1\frac{dx}{1+x^2}\end{align}$$
$$\text{as }\lim_{n \to \infty} \frac1n\sum_{r=1}^n f\left(\frac rn\right)=\int_0^1f(x)dx$$