let $a_k=\dfrac{1}{k\ln k}$
I found out that $a_{n+1} < a_n$ for all $n$.
Also, $\displaystyle\lim_{n\to \infty} a_n = 0$.
So by leibniz alternating series test, I said $\displaystyle\sum_{k=2}^{\infty}\frac{(-1)^{k+1}}{k\ln k}$ converges.
Is that true? Does it Absolute converge?