Proving/Disproving Series Transformation Properties

In summary, the conversation discusses whether there exists series \sum a_n and \sum b_n that meet the following conditions: 1) b_n can be obtained by rearranging the elements of a_n, 2) \sum b_n = 2 + \sum a_n, and 3) \sum |b_n| = 2 \sum a_n, where all series converge to finite values. After some discussion, it is concluded that it is impossible for all series to converge because rearranging the elements in an absolutely converging series does not change their value. The conversation also explores the idea of splitting the series into positive and negative "sub-series."
  • #1
daniel_i_l
Gold Member
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Homework Statement


Prove or disprove:
There exist series [tex]\sum a_n[/tex] and [tex]\sum b_n[/tex] so that:
1) you can get [tex] b_n [/tex] by rearranging the elements of [tex] a_n [/tex]
2) [tex]\sum b_n = 2 + \sum a_n [/tex]
3) [tex]\sum |b_n| = 2 \sum a_n [/tex]
(all the series converge to finate values)

Homework Equations





The Attempt at a Solution


From (1) I know that [tex]\sum |b_n| = \sum |a_n| [/tex] but I can't see how can to continue from here, can someone point me in the right direction?
Thanks.
 
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  • #2
You can't say that all series mentioned converge, because if [tex]\sum |b_n|[/tex] converges then the series is absolutely convergent which means that any reordering of the original series [tex]\sum b_n[/tex] converges to the same value, which implies that [tex]\sum a_n = \sum b_n[/tex] by virtue of (1). That means that (1) contradicts (2). So I have to say that it's impossible. But I could be wrong.
 
  • #3
DavidWhitbeck said:
You can't say that all series mentioned converge, because if [tex]\sum |b_n|[/tex] converges then the series is absolutely convergent which means that any reordering of the original series [tex]\sum b_n[/tex] converges to the same value, which implies that [tex]\sum a_n = \sum b_n[/tex] by virtue of (1).
How do you know that rearranging the elements in an absolutly converging series doesn't change their value?

EDIT: Oh, it's easy to see that that's true if you split up each of the series into positive and negative "sub-series".

Thanks for your help.
 
Last edited:

1. How do you prove/disprove a series transformation property?

To prove a series transformation property, you must show that it holds true for all values within a given range. This can be done through mathematical proofs or by providing a counterexample to disprove the property. It is important to carefully define the terms and assumptions of the property before attempting to prove or disprove it.

2. What is the significance of proving/disproving series transformation properties?

Proving/disproving series transformation properties is crucial for understanding the behavior of series and their transformations. It allows us to make accurate predictions and draw conclusions about the convergence or divergence of a series. Additionally, these properties serve as the foundation for more complex mathematical concepts and applications.

3. Can a series transformation property be both proven and disproven?

No, a series transformation property can only be either proven or disproven. If a property is proven, it means that it holds true for all values within a given range. If it is disproven, it means that there exists at least one counterexample that does not satisfy the property.

4. Are there any common methods or techniques for proving/disproving series transformation properties?

Yes, there are a few common methods and techniques that can be used to prove or disprove series transformation properties. These include using mathematical induction, direct proof, proof by contradiction, and proof by contrapositive. The method used will depend on the specific property being investigated.

5. Can a series transformation property be proven/disproved using numerical evidence?

No, numerical evidence alone is not enough to prove or disprove a series transformation property. While numerical evidence can provide supporting data, it is not considered a rigorous proof. A property must be proven using mathematical reasoning and logical arguments in order to be accepted as true.

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