Test the absolute convergence of the series $\sum_{k=1}^{\infty}(-1)^k\frac{\ln(k)}{k}$

Question

I need to test the following sequence for absolute convergence:

$$\sum_{k=1}^{\infty}(-1)^k\frac{\ln(k)}{k}$$

but I think I'm missing something. My comparison approach would be:

1. Disregard $(-1)^k$ since I need to show absolute convergence (?)
2. Denote that $\frac{\ln(k)}{k} > \frac{1}{k}$
3. Since $\frac{1}{k}$ is the harmonic series, it diverges, thus any sequence larger must also diverge
4. Thus, any series containing this sequence diverges, thus $\sum_{k=1}^{\infty}\frac{\ln(k)}{k}$ diverges and does not absolutely converge (I'm unsure if I can make this implication from the steps above)

I'm really mostly unsure if steps 1-3 are enough to infer 4), or if there are any better (simple) approaches to proving this?

Thank you guys very much in advance!