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.

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- November 4th 2012, 02:22 AMxinglongdadaSeries 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. - November 4th 2012, 03:43 AMgirdavRe: Series converge?
Use Abel transform.

- November 4th 2012, 03:59 AMxinglongdadaRe: Series converge?
Would you please be in detail?

- November 4th 2012, 04:47 AMgirdavRe: Series converge?
Sorry, it was a misleading hint. You can use Hölder's inequality to see it works if .

- November 4th 2012, 06:13 AMhollywoodRe: Series converge?
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