I came across with the following problem and I don't know how to solve it, so I need some help or any hints/steps of the solution.
problem: Find the limit of the sequence $\{a_n\}$, $$a_n= \left \{ \frac{n^2}{n^3+1} + \frac{n^2}{n^3+2} + \cdots + \frac{n^2}{n^3+n} \right \}$$
my thoughts
usually when the limit involves sums of such form I try to find the "hidden" Riemann sum.
so the first step, for me, is to wright
$$a_n= \frac{1}{n} \cdot \sum_{k=1}^{n} \frac{1}{1+\frac{k}{n^3}} $$
,but this seems to go nowhere
Thank you for your time.
By a simple majoration, you get that
$$n\frac{n^2}{n^3 +n} < a_n < n\frac{n^2}{n^3 +1}$$
$$\frac{1}{1 + \frac{1}{n^2} } < a_n < \frac{1}{1 + \frac{1}{n^3} } $$
And by squeeze theorem, the limit of $a_n$ is $1$