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Sum of the first n natural numbers is n(n+1)/2

  1. Sep 21, 2003 #1
    We know that the sum of the first n natural numbers is n(n+1)/2

    Can we express the product of the first n natural numbers without using the factorial symbol?
    It is possible to write a factorial as a sum. Any idea what it would look like?
     
    Last edited by a moderator: Feb 6, 2013
  2. jcsd
  3. Sep 21, 2003 #2

    Hurkyl

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    To my knowledge, there isn't any convenient way.

    You can do a "cheap" conversion from a product to a sum, though:

    ln Πf(n) = Σln f(n)

    So for factorials:

    ln(n!) = Σln n
    or
    n! = eΣln n
     
  4. Sep 22, 2003 #3

    ahrkron

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    Another way to write a factorial as a sum (which, I admit, sounds like cheating) is to use the Gamma function.

    n! = Gamma(n+1) = Integral(tx-1e-t)dt

    (the integral goes from zero to infinity)

    Since an integral is the limit of a sum, it is kinda what you wanted.
     
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