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Riemann Rearrangement Theorem

  1. Oct 13, 2012 #1
    1. The problem statement, all variables and given/known data
    I'm trying to compute the sum of the following series:

    S=1+[itex]\frac{1}{4}[/itex]-[itex]\frac{1}{16}[/itex]-[itex]\frac{1}{64}[/itex]+[itex]\frac{1}{256}[/itex]


    2. Relevant equations



    3. The attempt at a solution
    I'm not really sure how to begin this one. I know it probably involves Riemann's Rearrangement Theorem since this series is absolutely convergent.
     
  2. jcsd
  3. Oct 13, 2012 #2

    Ray Vickson

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    Are the successive signs really ++--+? What happens after that? Are you sure you copied the question correctly?

    RGV
     
  4. Oct 13, 2012 #3
    My apologies; I should have been clearer in my original post.

    The signs are ++-- ++-- ...
     
  5. Oct 14, 2012 #4
    Any help?
     
  6. Oct 14, 2012 #5
    Okay, so I think that if you think about it as the sum of two sums, that will help...

    think of the first sum as the sum of every odd indexed term, and the second sum as the sum of every even indexed term.

    S1 = 1-1/16+1/256.....
    S2 = 1/4-1/64+1/(16*64)....

    thus, the first sum will be

    S1 = [itex]\sum[/itex][itex](-1/16)[/itex]n from n = 0 to ∞.

    and the second, I'll let you figure out. but I think that this should help. (note, the second one needs a constant out front.

    hope this helps!
     
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