From the proofs of the Root and Ratio tests for a series, one deduces that if one of these tests shows divergence, then the terms of the series in question do not tend to zero.
I am therefore interested in finding an example of a divergent series (accessible to Calc II students) for which the Ratio or Root test is substantially easier to apply that the $n^{\rm th}$-term test (the Divergence Test). Does anyone know of one?
Thank you for any help, and I apologize in advance for the vague requirement ``substantially easier''.