Do you remember that when you started this thread, you insisted that you would have to use the Cauchy criterion? We haven't had any use for it in I(a), II(a) or II(b), so if you were right from the start, then we have to use it now. I told you what it means to use the Cauchy criterion, and how you should start, in post #64:
Fredrik said:
Definition: A series is convergent if and only if its sequence of partial sums is convergent. If the sequence is convergent, its limit is called the sum of the series.
Theorem: A series with real terms is convergent if and only if its sequence of partial sums is a Cauchy sequence.
You need to use this theorem to prove that your series is convergent. So the first thing you should write down is exactly what it means for your sequence of partial sums to be a Cauchy sequence.
Since then, I have told you at least twice that you need to read this and do what I'm saying at the end. I guess you're disappointed that no one else showed up, but if someone had, I would have wanted them to tell you nothing until after you have done this first step. They would have been wrong to do anything else.
Note that I told you that this is what it means to use the Cauchy criterion in post #5, 32 hours ago. I said "I assume", but you yourself confirmed that I was right in #6, half an hour later. Posts #9 (by micromass) and #25 (by me) also told you that I(b) is where you have to use the Cauchy criterion. #64 just spelled it out in as much detail as possible without violating forum rules.
One thing I didn't see from the start (this is the sort of thing you don't see until you actually start working on the problem) is that your teacher isn't just forcing you to use the Cauchy criterion so that you will have to learn it. He's doing it because if you just prove that \sum_{k=0}^\infty ar^k=\frac{a}{1-r} when |r|<1, you don't know if the series fails to converge for other values of r, or if the value of a is relevant when |r|≥1.
If you start by writing down what it means for the sequence of partial sums to be a Cauchy sequence, the rest is fairly straightforward. Keep in mind that a sequence of real numbers is convergent if and only if it's a Cauchy sequence. That's all I can tell you until you have actually done that first step.
