How to determine if this alternating series $\sum_{k=1}^\infty\frac{(-1)^k\ln k}k$ converges absolutely or conditionally?

Question

I'm trying to determine if the series $\sum\limits_{k=1}^\infty\frac{(-1)^k\ln k }k$ is converging absolutely or conditionally.

I've tried the ratio test and removed the alternating part i.e. $(-1)^k$ and then replaced every $k$ with $k+1$ and I got this: $$\frac{\dfrac{\ln(k+1)}{k+1}}{\dfrac{\ln k }k}.$$ I now don't know how to proceed with solving this or if this is the right way to solve it.

Answer 1

HINT: $\ln k>1$ for $k\ge 3$, so $\frac{\ln k}k>\frac1k$ for $k\ge 3$.