Answer
diverges
Work Step by Step
Given $$a_n=\sum_{n=1}^{\infty} \frac{n^{2}-4 n^{3 / 2}}{n^{3}}$$
Compare with $b_n=\sum \frac{1}{n} $, which is a divergent series with $p=1$. The by the limit comparison test:
\begin{align*}
\lim_{n\to \infty} \frac{a_n}{b_n}&= \lim_{n\to \infty}\frac{n^{2}-4 n^{3 / 2}}{n^{3}}\frac{n}{1}\\
&=1
\end{align*}
Thus, the given series also diverges.
