Does Non-Absolutely Convergent Series Imply Convergence of Product Series?

[arcadiaz04]

The problem states:
Suppose \sum a_n and \sum b_n are non-absolutely convergent. Show that it does not follow that the series \sum a_n b_n is convergent.

I tried supposing that the series \sum a_n b_n 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 \sum a_n b_n to be convergent. I get stuck again.

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

Is it true that \sum |a_n b_n| < \sum |a_n| \sum |b_n| ? 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

 

[AKG]

The problem states:
Suppose \sum a_n and \sum b_n are non-absolutely convergent. Show that it does not follow that the series \sum a_n b_n is convergent.

I tried supposing that the series \sum a_n b_n does converge, to find some contradiction.

Why would you do that? Do you know what you're being asked to prove?

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

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

Yes, it's obviously true (although it should be <, not <), and no, obviously it doesn't help.

Sorry it looks like I don't have much work done, but I've been looking at this for several days.

You need to know what you're trying to prove first. Basically, you want to find an example of series \sum a_n and \sum b_n such that:

a) both converge
b) neither converge absolutely
c) \sum a_nb_n doesn't converge

What types of series converge but don't converge absolutely? Ones that have some positive terms and some negative terms. Hint: take a_n = b_n. Then once you prove \sum a_n converges non-absolutely, you've automatically proven that \sum b_n converges non-absolutely. Moreover, if you do this, then you get:

\sum a_nb_n = \sum a_n^2

a sum of positive numbers. So whereas \sum a_n is supposed to be a series that converges, but doesn't converge absolutely, hence converges only because it has negative terms "balancing out" its positive terms, the series \sum a_n^2 has no negative terms, so it's "more likely" to be divergent. What's a very common example of a divergent series?

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.

You don't need any of that information.

 

[HallsofIvy]

AKG's point is that assuming \sum a_n b_n does converge would be perfectly reasonable if you were trying to prove that, under the given hypotheses, \sum a_n b_n never converged. But that is not the case. You want to show that the statement "If \sum a_n and \sum b_n converge then \sum a_n b_n" is NOT true. You want to find a counter example.

 

[arcadiaz04]

Have you both never heard of a proof by contradiction??

A statement can't be proven true by example.

 

[AKG]

Have you both never heard of a proof by contradiction??

A statement can't be proven true by example.

See here.
additional characters

 

[HallsofIvy]

Have you both never heard of a proof by contradiction??

A statement can't be proven true by example.

Did you not READ what we both said? The original problem was "Suppose \Sum a_n and \Sum bn are non-absolutely convergent. Show that it does not follow that the series \Sum a_nb_n is convergent." It does NOT follow. In other words prove that it is not true. You certainly can use a counter-example to prove something is NOT true.

 

[Office_Shredder]

Arcadiaz, prove that a natural number N is not necessarily even.

If you cite 3 as an example that isn't even, you've shown that N is not necessarily even.