I need to somehow figure out what happens with the following sum:
$\sum_{i=1}^{n^2 - 1} \frac{i^2}{[\frac{n^3}{3}]^2}$
when $n \rightarrow \infty$.
Should it be zero? Should it be a constant?
If I try and guess that sum of squares from $1$ to $n$ is not more than $n^3%$, it follows that a sum of squares from $1$ to $n^2 - 1$ is not more than $n^6$ (am I right with that?), but that gives me the same asymptotic order in the numerator is the same as in the denominator.
So I am in doubt if this is the right upper bound, maybe it's really less than $n^6$, but I'm not sure.