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Homework Help: Power Series Representation

  1. Jul 8, 2010 #1
    1. The problem statement, all variables and given/known data
    F(x)=∫(0 to x) tan^(-1)t dt. f(x)= infinite series ∑n=1 (-1)^(en)(an)x^(pn)?
    en=?
    an=?
    pn=?
    I know en = n-1


    2. Relevant equations



    3. The attempt at a solution
    Start with the geometric series
    1/(1 - t) = ∑(n=0 to ∞) t^n.

    Let t = -x^2:
    1/(1 + x^2) = ∑(n=0 to ∞) (-1)^n * x^(2n).

    Integrate both sides from 0 to x:
    arctan x = ∑(n=0 to ∞) (-1)^n * x^(2n+1)/(2n+1).

    Now that we have a series for arctan x...
    f(x) = ∫(0 to x) arctan t dt
    = ∫(0 to x) [∑(n=0 to ∞) (-1)^n * t^(2n+1)/(2n+1)] dt
    = ∑(n=0 to ∞) (-1)^n * [∫(0 to x) t^(2n+1)/(2n+1) dt]
    = ∑(n=0 to ∞) (-1)^n * x^(2n+2)/[(2n+1)(2n+2)].

    Shifting indices up one unit, we have
    f(x) = ∑(n=1 to ∞) {(-1)^(n-1)/[(2n-1)(2n)]} x^(2n).


    This gives me
    en = n-1
    an=1/(2n(2n-1))
    pn = 2n

    however an and pn are wrong. where am i going wrong?
     
  2. jcsd
  3. Jul 8, 2010 #2

    vela

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    Your work looks okay to me, and your answer matches what Mathematica spits out. Why do you think it's wrong?
     
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