Divergent, convergent series

Question

Let $p$, $q \in \mathbb{R}$ and see the series $$ \sum_{n=2}^{\infty} \frac{1}{n^p(\ln n)^q} $$ View with the comparison criterion that if $p> 1$ then the series is convergent for all $q$, and if $p < 1$, it is divergent for all $q$.

Can anyone help me getting started?

Answer 1

Hint: We have
$$\sum_{n=2}^{\infty} \frac{1}{n^{\alpha}} <\infty \Leftrightarrow \alpha > 1$$ and $$\ln(n) < n \forall ~ n \geq 1$$

Answer 2

Note that $$(\log n)^{q} = o(n^{p})$$ as $n \to \infty$ for all $p, q > 0.$