Answer
Converges
Work Step by Step
Consider $a_n=\dfrac{1+\cos n}{n^2}$
We know that $-1 \leq \cos^2 n \leq 1$
Now, $a_n \leq \dfrac{2}{n^2}$
and $\Sigma_{n=1}^\infty \dfrac{2}{n^2}$ is a convergent p-series with $p=2$
Hence, the series converges due to the direct comparison test.