limit of $n\sum_{k=1}^n\frac1{n^2+k^2}$ as n tends to infinity

Question

I tried to divide the the inside by n^2 and make into a Riemann sum form where my delta x is 1/n and my Xi* is i/n. However, I kind of get stuck after this and not sure how to carry on.

Answer 1

You may write, as $n \to \infty$, using a Riemann sum, $$ n\sum_{k=1}^n\frac1{n^2+k^2}= \frac1n\sum_{k=1}^n\frac1{1+\frac{k^2}{n^2}} \to \int_0^1\frac1{1+x^2}dx=\arctan (1) =\frac{\pi}4. $$