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Math Help - Power Series

  1. #1
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    Power Series

    a)Expand f(x) = (x+x^2)/(1-x)^3 as a power series.

    b)Use part a) to find the sum of the series: sum of (n^2)/(2^n) with n = 1 to infinity.

    Can someone help me with these? Thanks a lot.
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  2. #2
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    Quote Originally Posted by xfyz View Post
    a)Expand f(x) = (x+x^2)/(1-x)^3 as a power series.

    b)Use part a) to find the sum of the series: sum of (n^2)/(2^n) with n = 1 to infinity.

    Can someone help me with these? Thanks a lot.
    Use a Maclaurin series (a Taylor series expanded about 0):
    f(x) = \frac{x + x^2}{(1 - x)^3}

    f^{\prime}(x) = \frac{x^2 + 4x + 1}{(1 - x)^4}

    f^{\prime \prime}(x) = -\frac{2(x^2 + 7x + 4)}{(1 - x)^5}

    f^{\prime \prime \prime}(x) = \frac{6(x^2 + 10x + 9)}{(1 - x)^6}

    etc.

    So
    f(x) \approx f(0) + \frac{1}{1!} f^{\prime}(0) \cdot x + \frac{1}{2!} f^{\prime \prime}(0) \cdot x^2 + \frac{1}{3!} f^{\prime}(0) \cdot x^3 + ~ ...

    f(x) \approx x + 4x^2 + 9x^3 + ~ ...

    Can you spot the pattern? Can you see how it relates to the sum you are trying to find?

    -Dan
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