Both the ratio test (with a few generous inequalities) and the root test (definitely the best bet) work here.
Also, there is a general order that I think of and that I told my students when they tested for convergence. It is by no means infallible. Here's what I would say, and in this order:
- Write out a few terms! Get a feel for the series.
- Make sure the limit of the terms goes to zero.
- If it's integrable, use the integral test.
- If it alternates/telescopes, try the appropriate alternating/telescoping series tests.
- If it has a factorial, use the ratio test.
- If it has things raised to nth powers (like this one), use the root test.
- Use comparison - whichever feels more natural (I think basic is easier to see than limit, but so it goes)
I repeat, this is not infallible. Comparison tests can be scary, and some series are brilliantly bounded using astounding combinations of tests and ingenuity.