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Math Help - Power Series, Function Representation

  1. #1
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    Power Series, Function Representation

    Hi there,
    I am trying to Verify that f(x)=\frac{1}{3+2x}
    has power represenation of the sum from n=0 to infinity of
    \sum_{n=0}^{\infty}\frac{(-1)^n2^n}{3^{n+1}}*x^n
    I am aware of the differentiation and integration rules of the series, will I need those to complete the problem?
    Any direction would be greatly helpful.
    Thank you!!
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  2. #2
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    Quote Originally Posted by matt.qmar View Post
    Hi there,
    I am trying to Verify that f(x)=\frac{1}{3+2x}
    has power represenation of the sum from n=0 to infinity of
    \sum_{n=0}^{\infty}\frac{(-1)^n2^n}{3^{n+1}}*x^n
    I am aware of the differentiation and integration rules of the series, will I need those to complete the problem?
    Any direction would be greatly helpful.
    Thank you!!
    Your series outputs f^n(0) correctly, but it seems you are missing the division by n! aren't you?
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  3. #3
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    Quote Originally Posted by matt.qmar View Post
    Hi there,
    I am trying to Verify that f(x)=\frac{1}{3+2x}
    has power represenation of the sum from n=0 to infinity of
    \sum_{n=0}^{\infty}\frac{(-1)^n2^n}{3^{n+1}}*x^n
    I am aware of the differentiation and integration rules of the series, will I need those to complete the problem?
    Any direction would be greatly helpful.
    Thank you!!
    \frac{1}{3+2x} = \frac{\frac{1}{3}}{1 - \left(-\frac{2x}{3}\right)} = \frac{1}{3} - \frac{2x}{3^2} + \frac{2^2x^2}{3^3} - \frac{2^3x^3}{3^4} + ...
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  4. #4
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    Skeeter's point is that the geometric series \sum_{n=0}^\infty ar^n= \frac{a}{1- r}. Since \frac{1}{3+ 2x}= \frac{\frac{1}{3}}{1- (-\frac{2}{3})}, this is the sum of a geometric series with a= \frac{1}{3}, r= -\frac{2}{3}.
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