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Convergence for a series: what is wrong with my method?

  1. May 4, 2012 #1
    Convergence for a series: what is wrong with my method??

    1. The problem statement, all variables and given/known data

    For the following series, write formulas for the sequence an, Sn, and Rn, and find the limits of the sequences as n→∞.

    2. Relevant equations

    Sn is the partial sum of the series.

    Rn is the remainder and is given by
    Rn = S - Sn,
    where
    lim(n→∞) Sn = S



    3. The attempt at a solution

    an is pretty straightforward...

    an = Ʃ(1→∞) (-1)n+1 (2n+1)n / [ (n) (n+1) ]

    and as n→∞, an → 0

    For Sn, I am getting:

    Sn = 3 / (1x2) - 5 / (2x3) + 7 / (3x4) - 9 /(4x5) + ....

    = 3 (1/1 - 1/2) - 5 (1/2 - 1/3) + 7(1/3 - 1/4) - 9 (1/4 - 1/5) + ...

    = 3 - 1/2 (3+5) + 1/3 (5+7) - 1/4 (7+9) + ...

    = 3 - 8/2 + 12/3 -16/4 + ....

    = 3 - 4 + 4 - 4 + ....

    = 3 + Ʃ(1→∞) (-1)nx4

    so as n→∞, does Sn approach 3 or -1, or neither?

    Also, what would Rn be?

    thank you so much!
     
  2. jcsd
  3. May 4, 2012 #2

    LCKurtz

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    Re: Convergence for a series: what is wrong with my method??

    That is not ##a_n##. What you have on the right is ##S##, assuming the series converges.
    ##a_n## is the nth term of the series$$
    a_n=\frac{-1^n(2n+1)^n}{n(n+1)}$$Unless you have mistyped the problem, I don't think you will find ##a_n\rightarrow 0##. In any case you haven't shown any argument.
    ##S_n## is the sum of the first ##n## terms. The expression for ##S_n## would never end with a "...". You can't calculate ##S## or ##R_n=S-S_n## unless the series converges. Go back to the beginning and make sure you copied the problem correctly and start by proving whether or not ##a_n\rightarrow 0##.
     
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