real analysis - Series convergence/divergence

I'm trying to figure out whether the following series diverges or converges by using D'Alemberts (quotientcriteria), Cauchy (integral- and rootcriteria) and Leibniz convergence test for alternating series as well as direct comparisson tests.

$$\sum_{k=2}^\infty \frac{1}{(ln(k!))^2}$$

I'm very unclear on how to parse this series. My guess is that I have to find a series which is smaller since my guess is that it does converge.

$$ln(k!)\geq ln(k) => \frac{1}{ln(k)}\geq \frac {1}{ln(k!)}$$

But after trying the quotient criteria:

$$\frac{(ln(k))^2}{(ln(k+1))^2}$$

I find it unclear how to continue.

Answer

Of course it converges. You just have to show that $\ln(k!)^2$ grows fast enough.

For example:
We have $k!\ge (k/2)^{k/2}$, so for $k\ge 6$,

$$\ln(k!)^2\ge \frac{k^2}4 \ln(k/2)^2\ge \frac{k^2}{4}$$

and $\sum\limits_{k=2}^\infty \frac1{k^2}$ converges, so by comparison your series also does.