Answer
Divergent
Work Step by Step
Given $$\sum_{n=1}^{\infty} \frac{n+1}{n \sqrt{n}}$$
Use the Limit $c$ Comparison Test with $a_n =\dfrac{n+1}{n \sqrt{n}}$ and $b_n=\dfrac{1}{\sqrt{n}}$
\begin{align*}
\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}&=\lim _{n \rightarrow \infty} \frac{\sqrt{n}(n+1)}{n \sqrt{n}}\\
&=\lim _{n \rightarrow \infty} \frac{ n^{3/2}/n^{3/2}+n^{1/2}/n^{3/2}}{n^{3/2} /n^{3/2}}\\
&=\lim _{n \rightarrow \infty} \frac{1+1/n }{1}\\
&=1
\end{align*}
Since the $p-$ series $p<1$, $\displaystyle \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$ is divergent , then $\displaystyle\sum_{n=1}^{\infty}\dfrac{n+1}{n \sqrt{n}}$ is also divergent