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Math Help - Forumula for the product of the first 'n' numbers?

  1. #1
    Member mfetch22's Avatar
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    Forumula for the product of the first 'n' numbers?

    I know that we are able to derive a formula for the sum of the first 'n' positive natural numbers, that sum being:

    \sum_{i=1}^n i=\frac{(n)(n+1)}{2}

    I know that we can derive this formula by writing out the sum as

    S =1+2+ \cdot \cdot \cdot \cdot + (n-1) + (n)

    and doing some algebraic manipulation. But, what if we have the following:

    \prod_{i=1}^n i = (1)(2)(3) \cdot \cdot \cdot \cdot (n-2)(n-1)(n)

    Is there a way in which we can manipulate the product expansion (simmilar to the way in which we can manipulate the summation expansion) which will cause us to arrive at a direct formula for the product of the first 'n' positive integers simmilar to

    \sum_{i=1}^n i=\frac{(n)(n+1)}{2}

    such that we'd have something like the following

    \prod_{i=1}^n = f(n)

    [p.s. also, I realize that n! would denote the function I am refering to, but that sort if "notation" is not what I am refering to here. I'm asking weather there exists an explicit forumula for the product which does not have to be written in the form (n)(n-1)(n-2)......(3)(2)(1) or any form that includes the ".....", but rather a precise formula simmilar to the one used for the sum of the first "n" positive integers.]
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  2. #2
    Moo
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    A Cute Angle Moo's Avatar
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    Why do you want to find something more complicated ? You can view it like n*(n-1)!
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  3. #3
    Senior Member roninpro's Avatar
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    Quote Originally Posted by mfetch22 View Post
    [p.s. also, I realize that n! would denote the function I am refering to, but that sort if "notation" is not what I am refering to here. I'm asking weather there exists an explicit forumula for the product which does not have to be written in the form (n)(n-1)(n-2)......(3)(2)(1) or any form that includes the ".....", but rather a precise formula simmilar to the one used for the sum of the first "n" positive integers.]
    I'm a little bit concerned that your question is not well-posed. What do you mean by "precise formula"?
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  4. #4
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    The reason why we define n! to be the product of the first n positive integers is precisely because there is no simple formula for that product!

    I suspect that the reason there is a nice formula for the sum but not the product is that the ordering is defined by addition (1, 1+ 1, 1+ 1+ 1, ...) and multiplication just does not "play nicely" with addition.
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  5. #5
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    It's worth mentioning Stirling's Approximation. The fact that people bother with this may help convince the OP that coming up with a "nice" formula for n! is a lofty goal indeed.
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  6. #6
    MHF Contributor chisigma's Avatar
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    The product of the 'first n numbers' with n>0 is the value of the 'factorial function'...

    \displaystyle z!= \int_{0}^{\infty} t^{z}\ e^{-t}\ dt (1)

    ... when z=n ...

    Kind regards

    \chi \sigma
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