1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook