1. ## root/ratio tests

i need an example of a series of positive real numbers which converges by the root test, but where the ratio test is not useful at all

it has to do done using the strict senses of root and ratio tests which involve only upper limits (limsup), not limits in the strong sense

i would be grateful for any hints

2. I think that this is the usual example.
$a_n = 2^{ - n - \left( { - 1} \right)^n }$

It can be found in INFINITE SERIES by James M. Hyslop

3. Any sequence which has zero terms in it cannot be studied by the ratio test. But not with the root test. The root test always works (not that it will say convergence but it will say a limit at least). It is superior to it.

4. yeah i think "positive" here means "greater than zero", not including zero, but thanks for the info, i actually didn't even think about that limitation of the ratio test

In order to be able to consider $\lim \left| \frac{a_{n+1}}{a_n}\right|$ means that we cannot have $a_n = 0$ because otherwise it is blasphemy against math. Thus, we implicity assume that the terms of the series (or sequence) are non-zero.
In order to be able to consider $\lim \left| \frac{a_{n+1}}{a_n}\right|$ means that we cannot have $a_n = 0$ because otherwise it is blasphemy against math. Thus, we implicity assume that the terms of the series (or sequence) are non-zero.