Finding the infinite sum using Leibniz Test

Question

I have been given a task in my previous lecture to determine whether the infinite sum; $$\sum_{n=1}^\infty \frac{\cos (\pi n)\ln n}{n}$$ Converges or diverges. My perspective on the problem is that we will have to use the Leibniz test as $$\forall n\in \mathbb[N],\cos(\pi n)=(-1)^n $$ So I thought that because $\cos(\pi n)$ is always negative we can apply the Leibniz test. I don't, however, understand how we are meant to prove the conditions whereby $a_n\geq 0$ or $a_n \geq a_{n+1}$
Any help would be greatly appreciated.

Answer 1

$a_n \rightarrow 0$ as $n \rightarrow \infty$ by L’Hopital’s rule.

$$|a_n| \geq |a_{n+1}| \iff \frac{\operatorname{ln}n}{n} \geq \frac{\operatorname{ln}n+1}{n+1} \iff n^{n+1} \geq (n+1)^n $$

Which holds for all $n$ large enough.