Question

Determine whether the sequence an=n312+n322++n3n2 converges or diverges. If it converges, find the limit.

Answer 1

Perhaps it suffices to say that

$\displaystyle\frac1{1^2}+\cdots+\frac1{n^2}=\sum_{k=1}^n\frac1{k^2}$

converges to $\frac{\pi^2}6$ as $n\to\infty$, while $n^3\to\infty$. Then

$\displaystyle\lim_{n\to\infty}a_n=\infty\cdot\frac{\pi^2}6=\infty$

so $a_n$ diverges.