Let a series be given as $$\sum_{m=2}^{\infty}\frac{\cos(\pi m)}{\ln(\ln(m))}$$
Is it converges conditionally, converges absolutely or diverges.
Attempt
I have found out that the series does not converge absolutely since the positive sequence that can be written as
$$\frac{1}{|\ln(\ln(m))|}\geq \frac{1}{n}, \quad n\geq2 $$
and since we know the harmonic series diverges then so must the positive series in question. Now, for convergence I am kind of lost. Using alternating series test it does appear to work since I need to argue that $$\frac{1}{|\ln(\ln(m))|},\quad m\geq2$$
is monotonically decreasing and that the limit is $0$ at infinity. The limit at infinity seems straight forward but that it is monotonically increasing for every $m\in\mathbb{N}$ is simply not true when graphing it.
*Another confusion is that if I am allowed to use my comparison with the harmonic since I consider $m \geq 2$ and not $m\geq 1$.
If you exclude the term with $m = 2$, the function $\frac{1}{|\ln(\ln(m))|}$ is monotonically decreasing for $m \geq 3$, so you can use the alternating series test. In general, you only need that the absolute values are monotonically decreasing after some point.
Regarding comparison with the harmonic series, it is again ok to exclude the first term, since a divergent series is still divergent if the first term is excluded.