Convergence $\sum_{n=2}^{\infty} { \frac{\sqrt[n]{n^{p}}}{n\ln{n}} }$

Question

Help please. I need to check the convergence $$\sum_{n=2}^{\infty} { \frac{\sqrt[n]{n^{p}}}{n\ln{n}} }$$Tried with Leibniz, but can't check monotony.

Answer 1

First we have

$$\frac{\sqrt[n]{n^p}}{n\log n}\sim_\infty \frac{1}{n\log n}$$ secondly the series $$\sum_n \frac{1}{n\log n}$$ is divergent using the integral test hence your series is divergent.