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Math Help - expansion of infinite series

  1. #1
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    expansion of infinite series

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

    If the expansion of f(x) in ascending power of x is

    c_0+c_1x+c_2x+c_3x+...+c_rx^r+...

    find c_0,c_1 and c_2 and show that c_3=11
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  2. #2
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    Quote Originally Posted by thereddevils View Post
     f(x)=\frac{x^2+5x}{(1+x)(1-x)^2}

    If the expansion of f(x) in ascending power of x is

    c_0+c_1x+c_2x+c_3x+...+c_rx^r+...

    find c_0,c_1 and c_2 and show that c_3=11
    First notice that

    \frac{x^2+5x}{(1+x)(1-x)^2} = \frac{3}{(1-x)^2} - \frac{2}{1-x} - \frac{1}{1+x}


    If you consider the following geometric power series

    \frac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 \cdots

    so

    \frac{1}{1+x} = 1 - x + x^2 - x^3 + x^4 - + \cdots


    \frac{d}{dx} \left( \frac{1}{1-x}\right) = \frac{1}{(1-x)^2} = 1 + 2 x + 3 x^2 + 4 x^3 + 5 x^4 \cdots

    Substitute above giving

    3\left( 1 + 2 x + 3 x^2 + 4 x^3\right) - 2 \left( 1 + x + x^2 + x^3\right) - \left( 1 - x + x^2 - x^3\right) \cdots
    =(3-2-1) + (6 - 2+1)x + (9 - 2-1)x^2 +(12 - 2 +1) x^3 \cdots
    =5x + 6x^2 + 11 x^3 \cdots
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