$\displaystyle \sum (ln k)^2/k$
use the root test or the ration test
Before using blindly any "test", simply look at what the summand looks like. It is positive. The logarithm grows slowly to $\displaystyle \infty$, hence $\displaystyle \frac{(\ln k)^2}{k}$ is a bit larger than $\displaystyle \frac{1}{k}$. But... wait! we know that $\displaystyle \sum_k \frac{1}{k}$ diverges! So we just have to write $\displaystyle \frac{(\ln k)^2}{k}\geq \frac{1}{k}$ when $\displaystyle k\geq 3$ to conclude: the series diverges.