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Sum of an infinite series?

  1. Apr 15, 2012 #1
    1. The problem statement, all variables and given/known data

    Ʃ 4/(n(n+2)) from n=1 to n=infinity

    2. Relevant equations



    3. The attempt at a solution

    I tried using partial fractions to get A/n + B/(n+2), and I solved for A and B to get A=2 and B=-2

    I tried summing them up, so everything would cancel except the first & last term, but nothing cancels.

    I'm not sure of any other methods for finding sums or if I'm not just using this one wrong.

    Any help?
     
  2. jcsd
  3. Apr 15, 2012 #2

    HallsofIvy

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    Nothing cancels? Yes, partial fractions gives the addend as 2/n- 2/(n+2).
    When n= 1, that is 2- 2/3.
    When n= 2, that is 1- 2/4.
    When n= 3, that is 2/3- 2/5.
    When n= 4, that is 2/4- 2/6.
    When n= 5, that is 2/5- 2/7.
    When n= 6, that is 2/6- 2/8.
    When n= 7, that is 2/7- 2/9
    etc.

    I see a lot cancelling!
     
  4. Apr 15, 2012 #3

    LCKurtz

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    You have the right idea. Try writing out the first 6 or so terms of$$
    \sum_1^\infty(\frac 2 n - \frac 2 {n+2})$$leaving in the parentheses and not simplifying as you go. You will see some terms that cancel in the middle. Once you do that write the first ##n## terms.

    [Edit]I see Halls types faster than I do.
     
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