Is my decision diagram for the root test in the most general form?

Question

I created a decision diagram for the root test:

File Decision diagram for the root test.svg of Stephan Kulla licensed under CC-BY-SA 4.0

Is there any case missing or is it possible to formulate one of the cases in a more general form? I want to make sure that after three "No's" there is really no way to apply the root test...

Answer 1

I think you can probably almost arbitrarly refine that diagram further at the expense of complexity. For example: if $(a_n)$ does not alternate and

$$\text{if } \sqrt[k]{|a_k|} \geq \sqrt[k]{\frac{1}{k}} \text{ for almost all } k \in \mathbb{N} \text{, then } \sum_{k=1}^\infty a_k \text{ diverges.}$$

In my opinion your diagram is sufficient since the target group seems to be students new to the subject. I think most books do not even cover the bottom case.