Series that sums to 1/pi.

  1. CarlB

    CarlB 1,204
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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.

  2. jcsd
  3. there's something for 2/pi, look at in pi formulas.
    there are also formulas 1/pi but i didn't see a contiued fraction there.
  4. CarlB

    CarlB 1,204
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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:

  5. hotvette

    hotvette 931
    Homework Helper

    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. 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. Office_Shredder

    Office_Shredder 4,487
    Staff Emeritus
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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. 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. disregardthat

    disregardthat 1,817
    Science Advisor

    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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