Non-absolutely convergent proof, help please

In summary: That's not what I'm trying to do. I'm trying to find a statement that allows \sum a_n b_n to be convergent.I don't know what you're trying to do.
  • #1
arcadiaz04
2
0
The problem states:
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
 
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  • #2
arcadiaz04 said:
The problem states:
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.
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 [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
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 [itex]\sum a_n[/itex] and [itex]\sum b_n[/itex] such that:

a) both converge
b) neither converge absolutely
c) [itex]\sum a_nb_n[/itex] 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 [itex]a_n = b_n[/itex]. Then once you prove [itex]\sum a_n[/itex] converges non-absolutely, you've automatically proven that [itex]\sum b_n[/itex] converges non-absolutely. Moreover, if you do this, then you get:

[tex]\sum a_nb_n = \sum a_n^2[/tex]

a sum of positive numbers. So whereas [itex]\sum a_n[/itex] 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 [itex]\sum a_n^2[/itex] 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.
 
  • #3
AKG's point is that assuming [itex]\sum a_n b_n[/itex] does converge would be perfectly reasonable if you were trying to prove that, under the given hypotheses, [itex]\sum a_n b_n[/itex] never converged. But that is not the case. You want to show that the statement "If [itex]\sum a_n[/itex] and [itex]\sum b_n[/itex] converge then [itex]\sum a_n b_n[/itex]" is NOT true. You want to find a counter example.
 
  • #4
Have you both never heard of a proof by contradiction??

A statement can't be proven true by example.
 
  • #5
arcadiaz04 said:
Have you both never heard of a proof by contradiction??

A statement can't be proven true by example.
See here.
additional characters
 
  • #6
arcadiaz04 said:
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 [itex]\Sum a_n[/itex] and [itex]\Sum bn[/itex] are non-absolutely convergent. Show that it does not follow that the series [itex]\Sum a_nb_n[/itex] 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
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.
 

1. What is a non-absolutely convergent proof?

A non-absolutely convergent proof is a type of mathematical proof that shows a series or sequence does not converge absolutely, meaning it does not converge when all terms are considered positive. This type of proof is often used in analysis and number theory.

2. Why is a non-absolutely convergent proof important?

Non-absolutely convergent proofs are important because they help us understand the behavior of series and sequences, which are crucial in many areas of mathematics and science. They also provide a way to show that a series or sequence does not converge, which can be useful in disproving conjectures or solving problems.

3. How is a non-absolutely convergent proof different from other types of proofs?

A non-absolutely convergent proof is different from other types of proofs, such as absolutely convergent proofs, because it does not rely on the absolute values of terms in a series or sequence. Instead, it considers the behavior of the series or sequence as a whole, without considering the signs of individual terms.

4. What are some common techniques used in non-absolutely convergent proofs?

Some common techniques used in non-absolutely convergent proofs include the comparison test, the ratio test, and the root test. These tests help determine the behavior of a series or sequence and can be used to show that it does not converge absolutely.

5. Can a non-absolutely convergent proof be used to prove convergence?

No, a non-absolutely convergent proof is used to show that a series or sequence does not converge absolutely. It cannot be used to prove convergence, as there are other types of proofs, such as absolutely convergent proofs, that are used for that purpose.

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