Evaluating $\lim_{n \to \infty}\left(\frac{n}{n^3+1}+\frac{2n}{n^3+2}+ \dots + \frac{n^2}{n^3+n}\right)$ 
Compute the limit:
$$\lim_{n \to \infty}\left(\frac{n}{n^3+1}+\frac{2n}{n^3+2}+ \dots + \frac{n^2}{n^3+n}\right)$$

I tried applying the sandwich rule, constructing an upper and lower bound for the sequence, but I can't get the bounds to have the same limit.
Hints?

We have $$\frac{n^2(n+1)}{2(n^3+n)}=\frac{n}{n^3 +n } +...+\frac{n^2}{n^3 +n }\leq \frac{n}{n^3 +1 } +...+\frac{n^2}{n^3 +n }\leq \frac{n}{n^3 +1 } +...+\frac{n^2}{n^3 +1 }=\frac{n^2(n+1)}{2(n^3+1)}$$
