Answer
Converges
Work Step by Step
Let $u_n=\dfrac{n+1}{n^2 \sqrt n}$
This can be re-written as: $u_n=\dfrac{n+1}{n^2 \sqrt n}=\dfrac{n}{(n)^{5/2}}+\dfrac{1}{(n)^{5/2}}$
Here, the first series $\Sigma_{n=1} ^\infty \dfrac{1}{(n)^{5/2}}$ is a convergent p-series with $p=\dfrac{5}{2} \gt 1$
Also, the second series $\Sigma_{n=1} ^\infty \dfrac{1}{(n)^{3/2}}$ is convergent p-series with $p=\dfrac{3}{2} \gt 1$
Thus, the given series converges by the p-series.
