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.

Answer 2

Or in general we can apply the $ n^{th}$ divergence test stated as If $$\lim_{n \to \infty} \frac{\ln {n}}{\sqrt{n}}$$ does not exist then the series $$\sum_{n=1}^\infty \frac{\ln{n}}{\sqrt{n}} $$ Diverges. This can be a genral test for the testing of series But in above case the limit exist and equal to $0$ hence this test is not compatible for the series