Riemann Sum?

It can be treated as a Riemann sum, but you have to be a little careful due to the upper limit on the sum not being a scalar multiple of $n$ . The way I like treating it, however, is by dominated convergence, as this allows us to completely ignore this complication.

That is, first define the functions $\begin{aligned} f(x) = \frac{1}{1+x^2} &, & f_n(x) = \begin{cases} f\left(\frac{\lceil{nx}\rceil}{n}\right) & 0 < x \leq n \\ 0 & \text{otherwise}\end{cases} \end{aligned}$

and then by simply writing out the integral of $f_n$ and applying the Dominated Convergence Theorem $\displaystyle\text{(note }0 \leq f_n \leq f = \lim_{n\to\infty} f_n\text{ on } (0,\infty) \text{),}$ $\lim_{n\to\infty} \sum_{k=1}^{n^2} \frac{n}{n^2+k^2} = \lim_{n\to\infty} \int_0^\infty f_n(x)\,dx = \int_0^\infty f(x)\,dx = \arctan(x)\Big|_0^\infty = \boxed{\frac{1}{2}\pi}$