Series converge?

Suppose the series converges, whether or not the series converge?

I think it does not necessarily converge, but I could not find a counterexample. Would you help me? Thank you.

Use Abel transform.

Would you please be in detail?

Sorry, it was a misleading hint. You can use Hölder's inequality to see it works if .

Yes, so a counterexample would have to have to converge conditionally.

I think you might be able to prove that it converges, though. If the first series converges, then . Can you use this to show that the second series converges?

- Hollywood