Answer
converges
Work Step by Step
Given $$\sum_{n=1}^{\infty} \frac{\cos n}{n^{3 / 2}}$$
We check $\sum_{n=1}^{\infty} |\frac{\cos n}{n^{3 / 2}}|$
Since
\begin{align*}
|\cos n | &\leq 1\\
\frac{|\cos n |}{n^{3/2}} &\leq \frac{1}{n^{3/2}}
\end{align*}
We know that $\sum \frac{1}{n^{3/2}}$ is a convergent $p\gt1$ series. Thus, $\sum_{n=1}^{\infty} |\frac{\cos n}{n^{3 / 2}}|$ also converges and hence the given series converges.
