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Series question

  • Thread starter daniel_i_l
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  • #1
daniel_i_l
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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.
 

Answers and Replies

  • #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
daniel_i_l
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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:

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