- #1

arcadiaz04

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Suppose [itex]\sum a_n[/itex] and [itex]\sum b_n[/itex] are non-absolutely convergent. Show that it does not follow that the series [itex]\sum a_n b_n[/itex] is convergent.

I tried supposing that the series [itex]\sum a_n b_n[/itex] does converge, to find some contradiction. So the series satisfies the cauchy criterion and the definition of convergence. I can't break the series apart (or can I?) so this is where I get stuck.

Then I wrote the implications of the first sentence to try to come up with a statement that doesn't allow [itex]\sum a_n b_n[/itex] to be convergent. I get stuck again.

What does a series being non-absolutely convergent imply that is useful?

Is it true that [itex]\sum |a_n b_n|[/itex] < [itex]\sum |a_n|[/itex] [itex]\sum |b_n|[/itex] ? I don't know if that would help

Sorry it looks like I don't have much work done, but I've been looking at this for several days.Note: The section in which the problem is assigned talks about the boundedness criterion for convergence, the Cauchy criterion for convergence, and absolute convergence, so I was hoping to come up with a proof that uses the information from the section.

Thanks

CD