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 $\displaystyle \sum a_k$ be a series with positive terms and suppose that $\displaystyle \frac{a_{k+1}}{a_k}\rightarrow\lambda$.

- If $\displaystyle \lambda<1$, then $\displaystyle \sum a_k$ converges. (this proof was given to us already)
- If $\displaystyle \lambda>1$, then $\displaystyle \sum a_k$ diverges.
- If $\displaystyle \lambda=1$, 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 $\displaystyle \sum a_k$ 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.