What is the interval of convergence for $\sum \frac {\ln^3 n}{n^p}$

Question

I need to find the right interval to make sum $\large{\frac{\ln^3n}{n^p}}$ converge. I guess that p>1, still I don’t know how to prove it. I tried using integrals and Taylor series, no success though.

Answer 1

Note that for $p>0$

$$\frac{\frac{\log^3 n}{n^p}}{\frac1{n^a}}\to0$$

for $0<a<p$ and $\sum \frac1{n^a}$ converges for $a>1$, then the given series converges by limit comparison test with $\sum \frac1{n^a}$ for $1<a<p$ and diverges for $p\le 1$.