I don't know, ratio seems to be of the chart here, root likewise. Have you tried the integral test?
I may be being a bit dense here, but what does the notation denote? Is this a fractionally iterated logarithm, a fractional derivative (not likely I would have thought), abused notation for , ... ?
By the way as grows more slowly than any positive power of , so does and in particular for large enough there exists a such that , and so the series diverges.
CB