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Math Help - the positive root

  1. #1
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    the positive root

    Prove that the positive root of


    x(x + 1)(x + 2) (x + 1981) = 1


    is less than 1/1981!
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  2. #2
    MHF Contributor red_dog's Avatar
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    Let f(x)=x(x+1)(x+2)\ldots (x+1981)-1
    f(0)=-1

    f\left(\frac{1}{1981}\right)=\frac{1}{1981}\left(1  +\frac{1}{1981}\right)\left(2+\frac{1}{1981}\right  )\ldots\left(1981+\frac{1}{1981}\right)-1=

    =\left(1+\frac{1}{1981}\right)\ldots\left(1+\frac{  1}{1981^2}\right)-1>0

    (the factors of the product are greater than 1, so the product is greater than 1)
    f is continuos, so exists x_0\in\left(0,\frac{1}{1981}\right) such that f(x_0)=0
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  3. #3
    Senior Member JaneBennet's Avatar
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    Red Dog, I believe Perash is writing \frac{1}{1981!}\ \color{white}\frac{.}{\color{black}^{\leftarrow\te  xtrm{factorial}}}
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