# Non-absolutely convergent proof, help please

1. Nov 29, 2006

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

2. Nov 30, 2006

### AKG

Why would you do that? Do you know what you're being asked to prove?
Yes, it's obviously true (although it should be <, not <), and no, obviously it doesn't help.
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?
You don't need any of that information.

3. Nov 30, 2006

### HallsofIvy

Staff Emeritus
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.

4. Nov 30, 2006

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

A statement can't be proven true by example.

5. Nov 30, 2006

### AKG

See here.

6. Nov 30, 2006

### HallsofIvy

Staff Emeritus
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.

7. Nov 30, 2006

### Office_Shredder

Staff Emeritus
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.