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Homework Help: Representing a function as a power series

  1. Apr 27, 2008 #1

    grothem

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    1. The problem statement, all variables and given/known data
    Evaluate the indefinite integral as a power series and find the radius of convergence

    [tex]\int\frac{x-arctan(x)}{x^3}[/tex]


    I have no idea where to start here. Should I just integrate it first?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
    grothem, Apr 27, 2008
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  3. Apr 27, 2008 #2

    Dick

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    Sure, you could do that. But, I think what they want to do is expand arctan(x) as a power series around 0 and then integrate.
     
    Dick, Apr 27, 2008
  4. Apr 28, 2008 #3

    grothem

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    ok. So arctan(x) = [tex]\int\frac{1}{1+x^2}[/tex]
    = [tex]\int\sum (x^(2*n))[/tex]
    = [tex]\sum\frac{x^(2(n+1)}{2(n+1)}[/tex]

    is this what you mean?
     
    grothem, Apr 28, 2008
  5. Apr 28, 2008 #4

    Dick

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    That's one way to get a series for arctan, yes. But you forgot a (-1)^n factor. The expansion of 1/(1-x) has all plus signs. 1/(1+x) doesn't.
     
    Dick, Apr 28, 2008
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