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Math Help - proof

  1. #1
    Junior Member
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    proof

    Can somebody check my solution please?

     <br />
1 - x/1! + x(x-1)/2! +..+(-1)^n x(x-1)(x-2)....(x-n+1)/n! <br />
     <br />
= (-1)^n x(x-1)(x-2)....(x-n)/n!<br />
    Proof: by using derivation I will show that 1,2,3,,,n are simple roots of both polynomials, that implies they must be equal
    Am I right? Thank you very much.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by sidi View Post
    Can somebody check my solution please?

     <br />
1 - x/1! + x(x-1)/2! +..+(-1)^n x(x-1)(x-2)....(x-n+1)/n! <br />
     <br />
= (-1)^n x(x-1)(x-2)....(x-n)/n!<br />
    Proof: by using derivation I will show that 1,2,3,,,n are simple roots of both polynomials, that implies they must be equal
    Am I right? Thank you very much.
    the RHS is incorrect! it should be (-1)^n(x-1)(x-2) \cdots (x-n)/n!. of course if you can prove that 1, 2, ... , n are the roots of the LHS, then you're done, because the degrees and the leading

    coefficients of both sides are equal. (what does this equation have anything to do with "advanced" algebra anyway??)

    Edit: on second thought, i think it's easier to prove the identity by induction (over n of course!).
    Last edited by NonCommAlg; May 3rd 2009 at 03:49 AM.
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