What is the Simplest Series that Sums to 1/pi?

[CarlB]

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

 

[MathematicalPhysicist]

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.

 

[CarlB]

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

 

[hotvette]

Some time ago I remember seeing an iterative method for calculating $\pi$ (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.

 

[Radon-ruler]

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!

 

[Office_Shredder]

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

 

[Radon-ruler]

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 :)

 

[disregardthat]

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.