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Series that sums to 1/pi.

  1. Sep 26, 2005 #1

    CarlB

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    If I want a series that sums to pi there are a lot of choices. I seem to recall that there is also at least one simple series that sums to a rational multiple of 1/pi, but I can't recall what it is.

    I managed to find a continued fraction expansion that gives 1/pi, but it didn't seem to produce a very simple infinite series.

    The motivation for this problem is that I've been working on a physics problem where the answer is "2/9", and one begins with "2 pi / 3". If there were a series that came to 1/pi or better yet 1/(3 pi), then I might be able to guess a physical process (i.e. a series of Feynman diagrams) that would give that sum. Anyone have any clues?

    [edit]Maybe that continued fraction expansion is what I'm looking for. Basically, it's a continued fraction expansion for pi, but when one eliminates the first term, one gets an expansion for 1/pi. This seems like the kind of thing that might show up in a resummation of Feynman diagrams.[/edit]

    No I am not in school, and this is not homework.

    Carl
     
  2. jcsd
  3. Sep 27, 2005 #2

    MathematicalPhysicist

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    there's something for 2/pi, look at mathworld.com in pi formulas.
    there are also formulas 1/pi but i didn't see a contiued fraction there.
     
  4. Sep 27, 2005 #3

    CarlB

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    Just what I needed. Now for some poking and hoping.

    By the way, their continued fraction expansions for Pi are here:
    http://mathworld.wolfram.com/PiContinuedFraction.html

    Carl
     
  5. Oct 2, 2005 #4

    hotvette

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    Some time ago I remember seeing an iterative method for calculating [itex]\pi[/itex] (may or may not be the same as the continued fraction solution). If anybody is interested, I'll see if I can dig it up.
     
  6. Jul 24, 2010 #5
    The sum I feel would be most suited to this project can be found on ramujan's wiki page in the adult hood section. Sorry I can't just paste it for you, I'm on my phone :)

    I also have some rough thoughts on how one might procede with the physical process. One place you might want to look is at the category#2 version of the fourier transform... which is almost one of those langlans program thing.

    It's a cool idea, good luck with it!
     
  7. Jul 24, 2010 #6

    Office_Shredder

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    Yes, the amazing five year quest to find a formula that is available on wikipedia.

    We can probably parlay this into a book deal, and maybe a movie deal also
     
  8. Jul 24, 2010 #7


    I think finding a series of feynman diagrams corresponding to that sum would be fun. Feel free to do something else if you disagree :)
     
  9. Jul 24, 2010 #8

    disregardthat

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    The taylor expansion of 1/(2 arcsin(x)) at 1 is an obvious alternative, but probably not easy to compute. There might be some problems concerning the behavior of the function which I have not looked into.
     
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