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Math Help - power series help

  1. #1
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    power series help

    my question is:
    Expand g(x) = (1 - 2x)^(-1) as a powers series in (x+2).

    I don't even know where to start, can anyone help me
    thanks in advance
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  2. #2
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    Quote Originally Posted by giygaskeptpraying View Post
    my question is:
    Expand g(x) = (1 - 2x)^(-1) as a powers series in (x+2).

    I don't even know where to start, can anyone help me
    thanks in advance

    \displaystyle{\frac{1}{1-2x}=\frac{1}{5-2(x+2)}=\frac{1}{5}\cdot\frac{1}{1-\frac{2}{5}(x+2)}=\frac{1}{5}\sum\limits^\infty_{n  =0}\frac{2^n}{5^n}(x+2)^n} , valid for \displaystyle{\left|\frac{2}{5}(x+2)\right|< 1

    Tonio
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  3. #3
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    What Tonio did was treat the fraction \frac{1}{1- \frac{2}{5}(x+ 2)} as " \frac{1}{1- r}", the sum of a geometric series with r= \frac{2}{5}(x+ 2). That is the simplest and best way to do this problem.

    You could also taken derivatives of f(x)= (1- 2x)^{-1} and use the formula for a Taylor's series at x= 2 or used the "generalized binomial theorem" to expand (a+ b)^{-1} with a= 1, b= -2x.
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