I think I have an answer that works, but I'm not 100% sure.
We know that diverges.
I believe that, for n > 2, it can be shown that . Thus diverges.
I tried it as a guess, and so I worked backwards from the desired result:
It seems like the last statement holds true for all n > 2, and the steps could simply be written in reverse order to prove what we wanted to prove.
To prove this last inequality, you can say that the right hand side converges toward . In particular, it is bounded, and hence less than for large .
Alternatively, from the beginning, one can write so if you know the comparison test using asymptotic equivalence, you can conclude using and the divergence of .