Answer
Convergent
Work Step by Step
$\Sigma_{k=1}^{\infty}\frac{k lnk }{(k+1)^{3}}$
We note that $\sqrt k \gt lnk$
Then $\frac{\sqrt k}{k^{2}}=\frac{1}{k^{3/2}}$
As a series $\Sigma_{k=1}^{\infty}\frac{1}{k^{3/2}}$ converges because it is a p-series with $p=\frac{3}{2}\gt 1$
