Answer
Converges
Work Step by Step
Consider $a_n=\dfrac{\cos^2 n}{n^{3/2}}$
When we increase the numerator, the value of the fraction will always increase and when we decrease the denominator, the value of the fraction will always increase.
We know that $-1 \leq \cos n \leq 1; 0 \leq \cos^2 n \leq 1$
Thus, we get $a_n \leq \dfrac{1}{n^{3/2}}$
Here, $\Sigma_{n=1}^\infty \dfrac{1}{n^{3/2}},p=\dfrac{3}{2} \gt 1$, a convergent p-series.
Thus the series converges by the comparison test.