Showing $\lim_{n\to\infty} \left( \frac{n}{n^2+1^2} + \frac{n}{n^2+2^2} + \cdots + \frac{n}{n^2+n^2} \right) = \frac{\pi}{4}$

Question

How could I go about proving the following limit:

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

Answer 1

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.