Q:

Proof of value of infinite series, using definition of the integral

How do I prove that $$\lim_{n \to \infty }n^2\left ( \frac{1}{(n^2+1)^2} +\frac{2}{(n^2+2^2)^2}+...+\frac{n}{(n^2+n^2)^2} \right )=\frac{1}{4}$$ using the definition of the integral?

A:

$$
n^2\sum_{k=1}^n\frac{k}{(n^2+k^2)^2}=n^2\sum_{k=1}^n\frac{k}{n^4(1+(k/n)^2)^2}=\frac{1}{n}\sum_{k=1}^n\frac{k/n}{(1+(k/n)^2)^2}\ ,
$$
which is a Riemann sum, yielding
$$
\int_0^1 dx \frac{x}{(1+x^2)^2}=1/4
$$
in the limit.

