Claim: The series converges.
The series converges if and only if for every , there exists some such that for all with , we have .
Let be arbitrary but fixed, and let . Then the sequence is convergent, and it necessarily follows that is Cauchy. That is, for all , there exists some such that for all , we have .
But if , we have
which shows, directly from the definition, that the series is convergent.
Is this satisfactory? Or is there a more simplistic approach that I should be taking?
1) Do you have to do this by definition, or have you already seen the comparison test?
2) If you want to do this by definition, then you didn't prove anything. You stated that is convergent, which means converges, but that is what you have to prove! What you said is "this series converges because it converges" - this is not a proof.
Well, I've stated that is convergent since m is finite and fixed. The rest of the "proof" was intended to show that this extends to the case in which n goes to infinity.
Originally Posted by Defunkt
I did try my hand at the comparison test, but wasn't very successful there either. I'll give it another shot and see what I come up with.
If m is finite and fixed then is just a number. It does not depend on n.
The easiest way to prove this is by using the comparison test. I assume you have proved that converges. Now,
converges iff converges, since they both have positive terms.
Wow. I literally just made the connection with that particular series before coming back to see your post. :)