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
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
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.
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!
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 :)
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.