Showing $\lim_{n\to\infty} \left( \frac{n}{n^2+1^2} + \frac{n}{n^2+2^2} + \cdots + \frac{n}{n^2+n^2} \right) = \frac{\pi}{4}$ How could I go about proving the following limit:
$$
\lim_{n\to\infty} \left( \frac{n}{n^2+1^2} + \frac{n}{n^2+2^2} + \cdots + \frac{n}{n^2+n^2} \right) = \frac{\pi}{4}
$$

Hint: Write each term individually as
$$\frac{n}{n^2 + k^2} = \frac{1}{n} \frac{1}{1 + (k/n)^2}$$
and try to recognize a Riemann sum.
