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Math Help - write a function as a power series

  1. #1
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    write a function as a power series

    f(x)=x/(9 + x^2)

    We know that 1/(1-x) = (sigma)(0 to infinity)x^n

    find f(x) and the interval of convergence.
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by sprinks13 View Post
    f(x)=x/(9 + x^2)

    We know that 1/(1-x) = (sigma)(0 to infinity)x^n

    find f(x) and the interval of convergence.
    I will give you a hint this is f(x)=x\cdot\int\frac{1}{3^2+x^2}

    and to find the interval of convergence just take \lim_{n \to {\infty}}\frac{a_{n+1}}{a_n}<1 and solve for x
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  3. #3
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    I have managed to solve it:

    {a_n}=\frac{(-1)^nx^{2n+1}}{9^{n+1}}<br />

    But I didn't use your hint. Now I am curious, how did you go about solving this?
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  4. #4
    Moo
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    Hello,

    The most direct way is to say that :

    \frac{x}{9+x^2}=\sum_{n \geq 0} a_n x^n

    And to solve for a_n


    Now your result... It's ok.

    What mathstud meant is to observe... You know the power series for \arctan x, whose derivative is \frac{1}{1+x^2}

    From here, you can note that your function is x multiplied to what looks like the derivative of an arctan.
    Once you've got your power series, derivate it
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