Answer
diverges
Work Step by Step
Given $$\sum_{n=1}^{\infty} \frac{n}{\sqrt{n^{3 / 2}+1}}$$
Compare with $\sum \frac{1}{n^{1/4}}$, a divergent series ($p\lt1$):
\begin{align*}
\lim_{n\to \infty } \frac{n}{\sqrt{n^{3 / 2}+1}}.n^{1/4}&=\lim_{n\to \infty } \frac{1}{\sqrt{1+1/n^{3/2}}}\\
&=1
\end{align*}
Hence, $\sum_{n=1}^{\infty} \frac{n}{\sqrt{n^{3 / 2}+1}}$ also diverges.
