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Math Help - irrational number?

  1. #1
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    irrational number?

    Let x=1/1!+1/1!2!+1/1!2!3!+... Prove that x is an irrational number.
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  2. #2
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    Quote Originally Posted by mathman7 View Post
    Let x=1/1!+1/1!2!+1/1!2!3!+... Prove that x is an irrational number.
    Hopefully someone will give you a more elegant soln than mine!

    Assume that x is rational so the integers N and M exist such that:

    x=N/M

    Therefore, x*(1!2!3!.....M!) must be an integer.

    But
    x*(1!2!3!.....M!)=(1/1!+1/1!2!+1/1!2!3!+...)*(1!2!3!.....M!)

    Now the first M terms of this are integers. Define S to be the integer that is the sum of the first M integers then:


    x*(1!2!3!.....M!)=S+(1/(M+1)!+1/(M+1)!(M+2)!+...)

    Now provided M > 1
    0<(1/(M+1)!+1/(M+1)!(M+2)!+...)<(1/2^2+1/3^2+...)=\pi^2/6-1<1

    Since this term is between 0 and 1 it is not an integer and we have a contradiction with our statement that N is an integer and hence x must be irrational.
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