# Series that sums to 1/pi.

1. Sep 26, 2005

### 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?

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. Sep 27, 2005

### 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.

3. Sep 27, 2005

### CarlB

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. Oct 2, 2005

### 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.

5. Jul 24, 2010

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. Jul 24, 2010

### Office_Shredder

Staff Emeritus
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. Jul 24, 2010