The question I've been given involves proving two of the three different scenarios of the Ratio Test. We've been given it as follows.

Let

be a series with positive terms and suppose that

.

- If , then converges. (this proof was given to us already)
- If , then diverges.
- If , then the test is inconclusive.

The second and third parts need to be proven.

I believe I've gotten a good proof for the second one, where

diverges, though I could use a second opinion.

The third one, where the test is inconclusive, however, escapes me in terms of a proof. I could surely use a hand with that one.