Convergence of series $\sum_{n=1}^\infty \frac {cos(n \pi)}{(n+1)ln(n+1)} $

Question

How can I figure out if this series converges absolutely?

$$ \sum_{n=1}^\infty \frac {\cos(n \pi)}{(n+1)\ln(n+1)} $$

The ratio and root test are both inconclusive (according to Wolfram Alpha).

Answer 1

hint

$$\cos (n\pi)=(-1)^n $$

using alternate series test , it converges but using comparison with integral, it is not absolutely convergent :

$$\lim_{X\to+\infty}\int_1^X\frac {dt}{(t+1)\ln (t+1)}=$$

$$\lim_{X\to+\infty}\Big [\ln (\ln (t+1))\Bigr]_1^X =+\infty$$

thus, the series $$\sum \frac {1}{(n+1)\ln (n+1)} $$ is divergent.

Answer 2

Hints:

1) $\cos n\pi = (-1)^n \quad\forall n\in \mathbb{N}$

2) $\lvert (-1)^n \rvert = 1$

3) $\frac{1}{(n+1)\log(n+1)} \sim \frac{1}{n\log n}$ (limit comparison test)