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Math Help - Power series

  1. #1
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    Power series

    Hi I'm having troubles with a question and was wondering if someone could guide me through it:
    Given two mystery functions called anna and bob. Both have power series representations
    that work for all x \epsilon R
    Anna(x) = \sum_{n=0}^{\infty }a_{n}x^{n}, bob(x)= \sum_{n=0}^{\infty }b_{n}x^{n}
    Furthermore, anna(0) = 0; bob(0) = 1 and we know that deriving bob gives anna and deriving
    anna gives bob.
    a) Find anna and bob, i.e. nd the coefficients a_{n} and b_{n} of their power series.
    b) Express the exponential function e^{x} and e^{-x} as a combination of anna and bob. Conversely,
    express anna and bob as a combination of e^{x} and e^{-x}
    Thanks in advance
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  2. #2
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    Re: Power series

    Quote Originally Posted by Dragonkiller View Post
    Hi I'm having troubles with a question and was wondering if someone could guide me through it:
    Given two mystery functions called anna and bob. Both have power series representations
    that work for all x \epsilon R
    Anna(x) = \sum_{n=0}^{\infty }a_{n}x^{n}, bob(x)= \sum_{n=0}^{\infty }b_{n}x^{n}
    Furthermore, anna(0) = 0; bob(0) = 1 and we know that deriving bob gives anna and deriving
    anna gives bob.
    a) Find anna and bob, i.e. nd the coefficients a_{n} and b_{n} of their power series.
    b) Express the exponential function e^{x} and e^{-x} as a combination of anna and bob. Conversely,
    express anna and bob as a combination of e^{x} and e^{-x}
    Thanks in advance
    Here is the important information: You know that differentiating Anna will give Bob, and differentiating Bob will give Anna. So differentiating Anna TWICE will get back to Anna. This gives us the differential equation

    \displaystyle \begin{align*} A'' &= A \\ A'' - A &= 0 \\ \textrm{Characteristic Equation: } m^2 - 1 &= 0 \\ m^2 &= 1 \\ m &= \pm 1 \\ \textrm{Therefore } A &= C_1 e^x + C_2 e^{-x} \end{align*}

    We also know that \displaystyle \begin{align*} A(0) = 0 \end{align*} and \displaystyle \begin{align*} A'(0) = 1 \end{align*}, so we get

    \displaystyle \begin{align*} 0 &= C_1 e^0 + C_2 e^{-0} \\ 0 &= C_1 + C_2 \\ \\ 1 &= C_1 e^0 - C_2 e^{-0} \\ 1 &= C_1 - C_2 \\ \\ C_1 = \frac{1}{2}, C_2 = -\frac{1}{2}  \end{align*}

    So therefore

    \displaystyle \begin{align*} A = \frac{1}{2}e^x - \frac{1}{2}e^{-x} \end{align*} and \displaystyle \begin{align*} B = \frac{1}{2}e^x + \frac{1}{2}e^{-x} \end{align*}

    Go from here.
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