This is a straightforward application of the limit comparison test.
Note that if then converges.
The proposition is as follows:
If for all and there is a such that exists and is finite, then converges.
I'm honestly not even sure what angle to come at this from, since I can't see what good knowing that limit exists does in trying to prove the convergence of some series. If someone could just give me a nudge in the right direction here I should be OK.
This is a straightforward application of the limit comparison test.
Note that if then converges.
Legitimately embarrassed about not having seen that. This problem had been talked up so much that it didn't even occur to me that it could be such a straightforward application of one of the simpler convergence tests. One thing though, when you say , you mean , right?
Thanks a lot.
Actually I just noticed, I don't think you can apply the limit comparison test, as technically isn't necessarily a positive series. The proposition says that , whereas a positive series requires that , so I think I'm back at square one.
edit: but I guess the Limit Comparison test doesn't strictly require it to be a positive series, just that ...
Can't shake the feeling that my math professor is messing with me with the " isn't necessarily a positive series!!" warning.