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Series question

  1. May 5, 2008 #1


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    1. The problem statement, all variables and given/known data
    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)
    2. Relevant equations

    3. 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?
  2. jcsd
  3. May 5, 2008 #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.
  4. May 5, 2008 #3


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    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: May 5, 2008
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