I tested the following series for convergence, but I'm not sure if all the steps I took are strictly legal. Thus I would like to ask you to take a look at them.
I use the following fact to construct a series that is less than the one above for a comparison test.
Now I conduct a root test for this series to test it for convergence.
At this point the exponent of the exponential function is the problem, which I solve by using l'H˘pital's rule. (I'm not exactly sure if I am allowed to do this here.)
Now I move this newly obtained exponent back into the original limit calculation.
Since the smaller series diverges, also the greater one has to diverge.
I would appreciate it, if someone could tell me if I am right.