The sequence $\{r_k\}$ consisting of positive numbers has extended limit if $r_k \to \infty$ or $\exists L \in \mathbb R : r_k \to L$.
Limit Comparison Test :
If $\sum_{k=1}^\infty a_k$ and $\sum_{k=1}^\infty b_k$ are two series with positive terms, and the sequence $\frac{a_k}{b_k}$ has extended limit $\alpha$ , then we have :
1. If $\sum_{k=1}^\infty b_k$ is convergence, and $0 \le \alpha \lt \infty$ , Then $\sum_{k=1}^ \infty a_k$ is convergence.
2. If $\sum_{k=1}^\infty b_k$ is divergence, and $0 \lt \alpha \le \infty$, Then $\sum_{k=1}^\infty a_k$ is divergence.
After reading the proof of this theorem, it seems that there is no need for $\{a_k/b_k\}$ to be convergence.
- Find some predicates for the sequence $\{a_k/b_k\}$ such that the first part of theorem is true when that predicates hold.
- Find some predicates like the first part of the question such that the second part of the theorem is true when that predicates hold.