Answer
Converges
Work Step by Step
Let us consider $a_n=\dfrac{n! \ln n}{n(n+2)!}$
Apply direct comparison test.
$\Sigma_{n=1}^\infty \dfrac{(n!) \ln (n)}{(n)(n+2)!} \leq \Sigma_{n=1}^\infty \dfrac{n}{(n)(n+1)(n+2)}\leq \Sigma_{n=1}^\infty \dfrac{n}{n^3}=\Sigma_{n=1}^\infty \dfrac{1}{n^2}$
Here, the series $\Sigma_{n=1}^\infty \dfrac{1}{n^2}$ shows a convergent $p$-series with $p=2 \gt 1$
Thus, the series converges by the direct comparison test.