Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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)?
    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
    pn = 2n

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


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    Your work looks okay to me, and your answer matches what Mathematica spits out. Why do you think it's wrong?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook