I want to know the difference between ratio test and root test.
That is, I want to know the series which is tested by ratio but not root.
Or tested by root but not root.
As you know neither the ratio nor the root test help with $p$-series, i.e., $\sum_{n=1}^\infty \frac{1}{n^p}$ $( p > 1 )$.
Nevertheless we have a difficulty in testing $\sum_{n=1}^\infty n^2 e^{-n}$ by integral test. Surely we know by experience that the series converges.
But to prove we may use ratio or root test.
For this convenience, I think that root and ratio test is powerful.
But I do not know the difference between them in testing : As far as I know, the convergent series that is tested by root can be tested by ratio and vice versa.
Question : Is there a convergent series that is tested by root and cannot be tested by ratio ? Or is there a convergent series that is tested by ratio and cannot be tested by root ?