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Math Help - Need help with power series representation of function

  1. #1
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    Need help with power series representation of function

    I need help with the following:
    Find the power series representation of the function defined by:
    F(x) = the integral from 0 to x of cos (t1/2/2) dt

    Thanks!!!
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  2. #2
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    Re: Need help with power series representation of function

    By the second fundamental theorem of calculus, we have \displaystyle \begin{align*} F'(x) = \cos{\left( \frac{x^{\frac{1}{2}}}{2} \right)} \end{align*}.

    If you find the power series for \displaystyle \begin{align*} F'(x) \end{align*} and integrate it, you will have the power series you need.
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    Re: Need help with power series representation of function

    Quote Originally Posted by kimsanders82 View Post
    I need help with the following:
    Find the power series representation of the function defined by:
    F(x) = the integral from 0 to x of cos (t1/2/2) dt

    Thanks!!!
    We should know that the Taylor series for cosine is

    \cos(y)=\sum_{n=0}^{\infty}\frac{(-1)^{n}y^{2n}}{(2n)!}

    Now let t=\frac{\sqrt{t}}{2}

    This gives

    \cos\left( \frac{\sqrt{t}}{2}\right)=\sum_{n=0}^{\infty}\frac  {(-1)^{n}t^{n}}{4^n(2n)!}


    Now just integrate both sides from zero to x to get

    \int_{0}^{x}\cos\left( \frac{\sqrt{t}}{2}\right)dt=\int_{0}^{x}\sum_{n=0}  ^{\infty}\frac{(-1)^{n}t^{n}}{4^n(2n)!} dt

    F(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{n+1}}{4^n(2n)!(n+1)}
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