Evaluating the limit $\lim_{n\to\infty} \left[ \frac{n}{n^2+1}+ \frac{n}{n^2+2^2} + \ldots + \frac{n}{n^2+n^2} \right]$ Evaluating the limit $\displaystyle \lim_{n\to\infty} \left[  \frac{n}{n^2+1}+ \frac{n}{n^2+2^2} + \ldots + \frac{n}{n^2+n^2} \right]$
I have a question about the following solution:
We may write it in the form:
$$ \frac{1}{n} \left[ \frac{1}{1+(\frac{1}{n})^2} + \ldots + \frac{1}{1+(\frac{n}{n})^2} \right] $$
Somehow I need to figure out that the limit is actually the Riemann sum of $\frac{1}{1 + x^2}$ on $[0,1]$ for $\pi = 0 < \frac{1}{n} < \ldots < \frac{n}{n}$.
Can you explain to me to reach this conclusion? 

The number you seek is
\begin{equation}
L
= \lim_{n\to\infty}\sum_{k=1}^n\frac{n}{n^2+k^2}
= \lim_{n\to\infty}\sum_{k=1}^n\frac{1}{1+\left(\frac{k}{n}\right)^2}\cdot\frac{1}{n}\tag{1}
\end{equation}
Now, letting
\begin{align*}
\Delta x &= \frac{1}{n} & x_k &= \frac{k}{n}
\end{align*}
allows us to rewrite (1) as
$$
L = \lim_{n\to\infty}\sum_{k=1}^n\frac{1}{1+x_k^2}\Delta x
$$
If you're familiar with the standard construction of the Riemann integral, you will recognize this as
$$
L=\int_0^1\frac{1}{1+x^2}\,dx
$$
