I've reached my wit's end with this problem. We have not been taught about Ratio Tests or anything involving Radius of Convergence, but we have a problem asking for it.

For what values of $p$ does the sum converge?
$$\sum\_{n=1}^\infty{\ln(n) \over{n^p}}$$

I notice that it is "similar" to

$$\sum\_{n=1}^\infty{\frac {1}{n^p}}$$

And we know from the Integral Test of Convergence that this series converges when $\lvert p\rvert > 1$. Does this have any connection to the answer? Because I also notice that (via WolframAlpha) that when $p=1$ the series diverges, but when $p=1.1$ it converges.