How to prove that sequence is divergent.

Question

I have a certain sequence $$a_n= \frac{(-1)^n n^3}{n^3+2n^2+1} .$$ I know this sequence is divergent, but I do not know how to prove it. I tried using a theorem that says: $$ \lim_{n\to \infty} |a_n| = 0 \Rightarrow \lim_{n\to \infty} a_n = 0.$$ But he says nothing if $ \lim_{n\to \infty} |a_n| = L$ (in this case $L= 1$). So I would like to know how you proved that sequence is divergent.

Answer 1

$$a_n=\frac{(-1)^n}{1+\frac{2}{n}+\frac{1}{n^3}}.$$ I think now it's clear.

Answer 2

Hint: Consider it when $n$ is odd and $n$ is even separately!