<s>[INST] Evaluating $ \lim\limits_{n\to\infty} \sum_{k=1}^{n^2} \frac{n}{n^2+k^2} $ How would you evaluate the following series?
$$\lim_{n\to\infty} \sum_{k=1}^{n^2} \frac{n}{n^2+k^2} $$
Thanks.
 [/INST] Here's another approach.
First, note that
$$\begin{eqnarray*}
\sum_{k=n^2+1}^\infty \frac{n}{n^2+k^2}
&<& \sum_{k=n^2+1}^\infty \frac{n}{k^2} \\
&\le& n\int_{n^2}^\infty \frac{dx}{x^2} \\
&=& \frac{1}{n}.
\end{eqnarray*}$$
We also need the partial fraction expansion of $\coth x$,
$$\begin{eqnarray*}
\coth x &=& \lim_{N\to\infty} \sum_{k=-N}^N \frac{1}{x-i k \pi} \\
&=& \frac{1}{x} + \sum_{k=1}^\infty \frac{2x}{x^2+k^2\pi^2}.
\end{eqnarray*}$$
Then we find 
$$\begin{eqnarray*}
\lim_{n\to\infty} \sum_{k=1}^{n^2} \frac{n}{n^2+k^2}
&=& \lim_{n\to\infty}\left(
\sum_{k=1}^\infty \frac{n}{n^2+k^2} - \sum_{k=n^2+1}^\infty \frac{n}{n^2+k^2}
\right) \\
&=& \lim_{n\to\infty} \sum_{k=1}^\infty \frac{n}{n^2+k^2} \\
&=& \lim_{n\to\infty} \left(\frac{\pi}{2}\coth n\pi - \frac{1}{2n}\right) \\
&=& \frac{\pi}{2}.
\end{eqnarray*}$$
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