Determine the convergence/divergence of $\sum_{n=1}^{\infty}\frac{\ln{n!}}{n^3}$

Question

Does the series$$\sum_{n=1}^{\infty}\frac{\ln{n!}}{n^3}$$ converge or diverge? I initially thought about using the ratio test but then I got the ratio is $1$, so the test is inconclusive here.

I was thinking maybe using the comparison test? But I am not too sure which series to compare with. Any help would be appreciated.

Answer 1

notice $\ln n! = \ln 1 + \ln 2 + \ln 3 + .. + \ln n $

Answer 2

You could use Raabe's test

Given $\sum_{n=1}^\infty\dfrac{\ln(n!)}{n^3}$

$$\lim_{n\rightarrow\infty} \left(n\left(\dfrac{\dfrac{\ln(n!)}{n^3}}{\dfrac{\ln((n+1)!)}{(n+1)^3}}-1\right)\right)=2$$

Since $L>1$ the series converges.