Answer
Diverges
Work Step by Step
We know that $\cos^2n>0$ for all $n\geq 1$.
Then,
$\frac{n^2+\cos^2n}{n^3}>\frac{n^2+0}{n^3}$
$\frac{n^2+\cos^2n}{n^3}>\frac{1}{n}$
for all $n\geq 1$.
Since the series $\sum_{n=1}^\infty \frac{1}{n}$ diverges, it follows by the Direct Comparison Test with $a_n=\frac{n^2+\cos^2n}{n^3}$ and $b_n=\frac{1}{n}$ that the series $\sum_{n=1}^\infty \frac{n^2+\cos^2n}{n^3}$ diverges.
