Two alternating series with logarithms

First series. $b_n=\frac1{\ln n}$ is positive, decreasing (as $\ln n$ increases) and tends to $0$: the series converges. Since $\ln n\le n$ we have $\frac1{\ln n}\ge\frac1n$, so $\sum\frac1{\ln n}$ diverges: the convergence is conditional.

Second series. Let $f(x)=\frac{\ln x}x$. Then $f'(x)=\frac{1-\ln x}{x^2}<0$ for $x>e$, so $\left(\frac{\ln n}n\right)_{n\ge3}$ is decreasing, and it tends to $0$. The series converges. Since $\frac{\ln n}n\ge\frac1n$ for $n\ge3$, the series of absolute values diverges: conditional convergence.

Remark. The alternating series test requires *eventual* monotonicity only: the first terms (here $n=1,2$) do not matter.