Converges or diverges $\sum_{n=1}^\infty \frac{\ln n}{\sqrt n}$

Question

I was trying to find if the series $\sum_{n=1}^\infty \frac{\ln n}{\sqrt n}$converges or diverges. First, I tried ratio test and got the limit as 1. I tried Limit Comparison Test's and I only got 0's and $\infty$'s. Then I tried using $n\geq \ln n$ for Direct Comparison Tests, but I could not find a result. Can you help me to see what am I missing?

Answer 1

Note that $\sum 1/n^p$ diverges for all $p<1$.

So, the summand is lower bounded by $1/\sqrt{n}$ and is non-negative, so by the comparison test to $\sum 1/\sqrt{n}$ it diverges.