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Power Series ArcTan?

  1. Mar 27, 2010 #1
    Power Series ArcTan???

    1. The problem statement, all variables and given/known data
    Let f be the function given by f(t) = 4/(1+t^2) and G be the function given by G(x)= Integral from 0 to x of f(t)dt.
    A) Find the first four nonzero terms and the general term for the power series expansion of f(t) about x=0.
    B) Find the first four nonzero terms and the general term for the power series expansion of G(t) about x=0.
    C) Find the interval of convergence of the power series in part (B). Show the analysis that leads to your conclusion.


    2. Relevant equations
    d/dtArctan(t)=1/(1+t^2)


    3. The attempt at a solution
    A) a=4, R=-t^2. f(t)=Sum from n=1 to infinity of 4 * (-1)^n * t^2n
    First four terms: -4t^2 + 4t^4 - 4t^6 + 4t^8

    B) Integral from 0 to x of 4/(1+t^2)dt = 4arctan(t) from 0 to x = 4arctan(x)

    Now I don't know where to go from here. I don't know how to write the power series for the antiderivative of the original power series, since it is not in the standard form of a power series. Can anybody help?
     
  2. jcsd
  3. Mar 27, 2010 #2

    HallsofIvy

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    Re: Power Series ArcTan???

    No. You have the formula right but when n= 0, 4(-1)^n t^2n is 4. The first four terms are 4- 4t^2+ 4t^4- 4t^6.

    Why is it "not in the standard form of a power series"?

    You have that [itex]4/(1+ t^2)= 4\sum_{n=0}^\infty (-1)^n t^{2n}[/itex] and can integrate "term by term".
     
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