Answer
Converges
Work Step by Step
Consider $a_n=\dfrac{n+1}{n^2 \sqrt n}$
or, $a_n=\dfrac{n+1}{n^2 \sqrt n}=\dfrac{n}{n^{5/2}}+\dfrac{1}{n^{5/2}}$
Now, $\Sigma_{n=1} ^\infty \dfrac{1}{n^{5/2}}$ is convergent due to the p-series with $p=\dfrac{5}{2} \gt 1$
and $\Sigma_{n=1} ^\infty \dfrac{1}{n^{3/2}}$ is convergent due to the p-series with $p=\dfrac{3}{2} \gt 1$
Hence, the the sum of the series converges due to the convergent p-series.
