At first the problem seems an easy application of a convergence criterion... may be however that it covers up an insidious trap ...
From...
http://en.wikipedia.org/wiki/Convergent_series
Root test: suppose that the terms of the sequence are non negative and that there exists r such that...
(1)
If the series converges, if the series diverges, if the root test is inconclusive...
It is not quite obvious the reason to have in (1) instead of . Anyway in our case is...
(2)
Now, if we want to apply the root test as explained by Wiky, the problem is that an such that doesn't exists ...
Merry Christmas from Italy