To test for convergence of a series, I would usually take the limit of the series with root test's power 1/n.

But in some cases, I was told that I could use the power of the coefficient of a power series. For instance, in this series

, its coefficient has a power of 2p. I could just take the limit of the coefficient,

. So 3/4 is the radius of convergence, which is correct. If I took the limit of the root test with power 1/n, the answer would be wrong.

But then in another series like this one:

, when I take the limit of its coefficient with its power 2n,

, the answer for the radius of convergence is wrong. The correct answer is 1/(4e). Apparently, I will only get the 1/(4e) as the answer if I took 1/n in my root test instead of 1/2n.

In this case, how would I know when I should take 1/n or 1/2n or even 1/(2n+1) in my root test for a series?

Thanks!