2. If , then for large n. Therefore, is divergent.
3. Take the p-series
for any p.
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.