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Re: product

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  • Dario Alpern
    ... This product is not very difficult to compute if you know about the function gamma (the extension of factorials to the complex domain). You are trying to
    Message 1 of 11 , Dec 5, 2005
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      --- In primenumbers@yahoogroups.com, "Mark Underwood"
      <mark.underwood@s...> wrote:
      >
      > --- In primenumbers@yahoogroups.com, "Jacques Tramu"
      > <jacques.tramu@e...> wrote:
      > >
      > > I have computed the product and found
      > > n= 200 000 000 : 0.9048530178
      > > n= 1 000 000 0000 : 0.9048042986
      > > n= 1 500 000 000 : 0.9047987573
      > >
      >
      > I was just averaging some figures around the n = 40,000 area and it
      > worked out to be around .9105. Looking a Jacque's findings the
      > product seems to be ever so slowly shrinking.
      >
      > Contrast this the product 1/2 * 4/3 * 5/6 * 8/7 * ....
      >
      > and this appears to be converging to .599070...
      >
      > Mark
      >

      This product is not very difficult to compute if you know about the
      function gamma (the extension of factorials to the complex domain).

      You are trying to find:

      k=inf (4k+1)(4k+4) k=inf 2
      Prod ------------ = Prod 1 - ------------
      k=0 (4k+2)(4k+3) k=0 (4k+2)(4k+3)

      This is clearly convergent. But what is the limit?

      k=n (4k+1)(4k+4) k=n (k+1/4)(k+1)
      Prod ------------ = Prod -------------- =
      k=0 (4k+2)(4k+3) k=0 (k+2/4)(k+3/4)

      gamma(k+5/4) gamma(k+2) gamma(2/4) gamma(3/4)
      = ------------------------- ---------------------
      gamma(k+6/4) gamma(k+7/4) gamma(1/4) gamma(1)


      = A(k) x B

      where A(k) is the first fraction and B the second.

      We are interested in the value of A(k) as k->inf.

      Fortunately it turns out that the limit is 1. This can be seen by
      using Stirling approximation.

      Since gamma(1) = 1 and gamma(1/2) = sqrt(pi) we finally get:

      k=inf (4k+1)(4k+4) sqrt(pi) * gamma(3/4)
      Prod ------------ = ----------------------
      k=0 (4k+2)(4k+3) gamma(1/4)

      This is about 0.5990701173677961037199612

      Best regards,

      Dario Alejandro Alpern
      Buenos Aires - Argentina
      http://www.alpertron.com.ar/ENGLISH.HTM
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