Evaluate the limit by expressing it as a definite integral:
\begin{equation}
\lim_{n \to \infty} \sum_{k=n+1}^{2n-1} \frac{n}{n^2+k^2}
\end{equation}
I'm really confused about tackling this. Although I know it is a Riemann sum, I don't really understand this and am hoping to seek a correct approach to this question. I tried using $t=1/n$ and replacing the limit by
\begin{equation}
\lim_{t \to 0+} \sum_{k=n+1}^{2n-1} \frac{n^2}{n^2+k^2}t
\end{equation}
but I don't know what to proceed next.
I would appreciate some help/tips/hints on how to solve this. Many thanks!
Best Answer
$\textbf{Hint}$: Rewrite then reindex the sum as follows
$$\sum_{k=n+1}^{2n-1}\frac{n}{n^2+k^2} = -\frac{1}{2n}+\sum_{k=n}^{2n-1}\frac{n}{n^2+k^2}= -\frac{1}{n}+\sum_{k=0}^{n-1}\frac{n}{n^2+(k+n)^2}$$
The first term vanishes in the limit leaving the last term
$$\sum_{k=0}^{n-1}\frac{1}{1+\left(\frac{k}{n}+1\right)^2}\cdot\frac{1}{n}$$
Can you continue from here?