MHB Does the Sequence Converge or Diverge?

[karush]

$\tiny{s4.11.1.26} \\$
$\text{ Determine whether the sequence converges or diverges. If it converges, find the limit.} \\$
$$\displaystyle a_n=\frac{(-1)^n n^3}{n^3+2n^2+1}$$
$\text{ divide every term by $n^3$}$
$$\displaystyle a_n=\frac{(-1)^n }{1+\frac{2}{n}+\frac{1}{n^3}}$$
$\text{ take the limit}$
$$\displaystyle \lim_{{n}\to{\infty}} a_n=1$$
$\text{suggestions?}$

 

[Evgeny.Makarov]

The limit of $a_n$ is not 1. If we replace $(-1)^n$ with 1, then indeed the limit is 1. But as it is, $a_n$ oscillate between numbers that are close to $-1$ and those that are close to $1$.

 

[karush]

so it diverges