How to determine whether a series converges when it contains ɛ >0?

Question

$$\sum_{n=2}^{∞} \frac{1}{n\times ln(n)^{1+\epsilon}}$$

With $\epsilon > 0$

I'm unsure how to prove convergence. I'm allowed to use $\lim_{n \to \infty}$ ln $(n)^{\epsilon} = \infty$ without proof.

So how would I prove convergence?

Answer 1

It is more appreciative if you post your try. This is my comment:

It is easy to check that $\sum_{n=1}^\infty \frac{\ln^{-1-\varepsilon} {(n)}}{n}$ converges whenever $\varepsilon >0$.

Answer 2

You can use Integral Test, which then you can apply $$\lim_{n \to \infty} (n)^{\epsilon} = \infty $$ to show that the integral converges.