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Compact form for an infinite multiplication

  1. Jun 16, 2012 #1
    Hi,
    In the middle of the article about Franciscus Vieta, here:
    http://en.wikipedia.org/wiki/Franciscus_Vieta

    I see an infinite product as an expression for Pi:
    2 * 2/2^(1/2) * 2/(2+(2^(1/2))^(1/2) * ...

    I was wondering, how this could be written in compact form using math notation please?
     
  2. jcsd
  3. Jun 16, 2012 #2

    SammyS

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    I don't think you can write [itex]\displaystyle 2\times\frac{2}{\sqrt{2}}\times\frac{2}{\sqrt{2+ \sqrt{2}}}\times\frac{2}{\sqrt{2+\sqrt{2+\sqrt{2}}}}\times\frac{2}{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}\times\cdots[/itex] in compact form.
     
  4. Jun 17, 2012 #3
    Indeed, I don't think any specific notation can help simplify this ... er ... "continued fraction."

    I would define the recursive sequence [itex]S_n[/itex] such that:

    [itex]
    S_0 = 1\\
    S_1 = \sqrt2\\
    S_{k+1} = \sqrt{2+S_k}\text{ where } k>0\\
    [/itex]

    and use that sequence and capital pi notation to shorten the equation for [itex]\pi[/itex] into the infinite product:

    [itex]
    \pi=\prod \limits_{i=0}^{\infty} \frac{2}{S_i}
    [/itex]

    But then again, this seems more convoluted than compact.
     
  5. Jun 17, 2012 #4
    Thank you all, I learned from your answers that the other way of expressing that is by using a recursive expression combined together with Pi (∏) notation and that there is no unique tool in math-notation for a compact form in this case.
     
  6. Jun 17, 2012 #5
    I don't know if you want this particular formula for pi or any infinite product will do.

    If another is acceptable Google Wallis product.
     
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