How do you calculate this limit $\lim_{n\to\infty}\sum_{k=1}^{n} \frac{k}{n^2+k^2}$? How to find the value of $\lim_{n\to\infty}S(n)$, where $S(n)$ is given by $$S(n)=\displaystyle\sum_{k=1}^{n} \dfrac{k}{n^2+k^2}$$
Wolfram alpha is unable to calculate it.
This is a question from a questions booklet, and the options for the answer are--
$\begin{align}
&A) \dfrac{\pi}{2} \\
&B) \log 2 \\
&C) \dfrac{\pi}{4} \\
&D) \dfrac{1}{2} \log 2
\end{align}$

Clearly, 
\begin{align}
\lim_{n\to\infty}\sum_{k=1}^n\frac{k}{n^2+k^2} &=\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\frac{\frac{k}{n}}{1+\frac{k^2}{n^2}}
\stackrel{\text{Riemann sum}}\longrightarrow \int_0^1 \frac{x\,dx}{1+x^2}=\left.\frac{1}{2}\log (1+x^2)\right|_0^1\\
&=\frac{1}{2}\log 2.
\end{align}
