Let $r,s \in \mathbb{R}$. Consider $$ f(r,s) = \sum_{n=1}^\infty \frac{1}{n^s \ln^r n}. $$
- $s, r \le 0 \implies f(r,s)$ diverges
- $s=0, r < 0 \implies f(r,s)$ diverges
- $s<0, r = 0 \implies f(r,s)$ diverges
I could figure out what would happen when s and r are negative and zero but what other cases are possible and how do you go about solving those cases?