$\frac{1}{n} \sum_{k=1}^n k^2/(n^2 + k^2)$ How can I compute the following sum exactly: $$\frac{1}{n} \sum_{k=1}^n k^2/(n^2 + k^2)?$$ (I can approximate it by a Riemann integral, but that's not my goal).
 A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
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\begin{align}
{1 \over n}\sum_{k = 1}^{n}{k^{2} \over n^{2} + k^{2}} & =
{1 \over n}\,\Re\sum_{k = 1}^{n}{k \over k + \ic n} =
1 + {1 \over n}\,\Re\pars{-\ic n\sum_{k = 0}^{n - 1}{1 \over k + 1 + \ic n}}
\\[5mm] & =
1 + \Im\sum_{k = 0}^{\infty}\pars{{1 \over k + 1 + \ic n} -
{1 \over k + n + 1 + \ic n}}
\\[5mm] & =
\bbx{1 + \Im\pars{H_{n + \ic n} - H_{\ic n}}}\qquad\pars{~H_{z}:\ Harmonic\ Number~}
\end{align}
