Series that sums to 1/pi.

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CarlB
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Main Question or Discussion Point

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
 

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  • #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.
 
  • #3
CarlB
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loop quantum gravity said:
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.
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
 
  • #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.
 
  • #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!
 
  • #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
 
  • #7
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


I think finding a series of feynman diagrams corresponding to that sum would be fun. Feel free to do something else if you disagree :)
 
  • #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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