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? 

$$\frac1n\sum_{k=1}^n\frac1{1+\left(\frac kn\right)^2}\xrightarrow[n\to\infty]{}\int_0^1\frac{dx}{1+x^2}\left(=\frac\pi4\;.\;\;Why?\right)$$
Just note that the above sum is 
$$\sum_{k=1}^n f\left(\frac kn\right)(x_{k+1}-x_k)\,,\,\,\text{with}\;\;f(x)=\frac1{1+x^2}\;$$
and the partition of the unit interval 
$$\left\{0<\frac1n<\frac2n\ldots<\frac nn=1\right\}\;,\;\;\text{and}\;\;c_k:=\frac kn$$
