I'm currently practicing problems for an upcoming test, but I'm not sure how to prove this series semi-convergence. This is the series:

$$\sum U\_n=\frac{\left(-1\right)^n}{\ln n}$$

My professor puts as a hint: $\ln n\le n$ for $n$ great enough, but I'm not sure on the correct way of stating this. If it was divergent, I would simply state that we know $\frac{1}{n}$ is divergent and $\frac{1}{\ln n}\ge \frac{1}{n}$ so $\frac{1}{\ln n}$ is divergent by comparison but I'm not sure the same works for semi-convergence. Would this comparison be valid in the case of a semi-convergent series?