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Math Help - Finding power series representations?

  1. #1
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    Finding power series representations?

    1/[(1+2x)^2]

    Hint: Differentiate, find a power series for the derivative, then integrate the power series

    This hint really doesn't make sense to me. Why would you differentiate it first? If anything I feel like you should integrate it first, in which case you would get -1/(2+4x) and you can find a power representation for this. Any advice?
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    Quote Originally Posted by sj9110 View Post
    1/[(1+2x)^2]

    Hint: Differentiate, find a power series for the derivative, then integrate the power series

    This hint really doesn't make sense to me. Why would you differentiate it first? If anything I feel like you should integrate it first, in which case you would get -1/(2+4x) and you can find a power representation for this. Any advice?
    I agree, the hint doesn't make sense. If you start with the power series

    \frac{1}{1+x} = 1 - x + x^2 - x^3 + \cdots

    diffferentiate

    \frac{-1}{(1+x)^2} = - 1 + 2x - 3x^2 + \cdots

    then negate

    \frac{1}{(1+x)^2} = 1 - 2x + 3x^2 + \cdots

    then replace x with 2x you'll get the desired power series.
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