A quick perusal of the wikipedia page for $\pi$ yields a large collection of known series for $\pi$. In particular, these series are hypergeometric in nature, and have large (but finite) radius of convergence. 


My first question is, what is the best known series for $\pi$? Namely, a series of the form above with the largest radius of convergence.


My second question is one asked by Herbert Wilf (<http://www.math.upenn.edu/~wilf/website/UnsolvedProblems.pdf>), which is to ask whether there exists a sequence $(a\_n)$ with $a\_{n+1}/a\_n$ a rational functional of $n$ for all $n$, and the function $f(z) = \displaystyle \sum\_{n=0}^\infty a\_n z^n$ is an entire function, and $f(1) = \pi$. Presumably, such a function is not known to exist yet. Can anyone give any recent works that advances our understanding on this problem, or give some insight as to why such a function is so difficult to find?


Thanks!



Fast series for pi
I like Andre Henriques' rephrasing. The Borwein, Bailey, Plouffe series, with $$a\_n={1\over16^n}\left({4\over8n+1}-{2\over8n+4}-{1\over8n+5}-{1\over8n+6}\right)$$ would have radius of convergence $r=16$. Bellard gives a more complicated one with $r=1024$. Pschill has one with 21 terms and $r=2^{30}$. If you'll accept $f(1)=1/\pi$, D and G Chudnovsky give $$a\_n={12\over\sqrt{640320^3}}(-1)^n{(6n)!\over(n!)^3(3n)!}{13591409+54514013n\over(640320^3)^n}$$ All of these are taken from Chapter 16 of Arndt and Haenel, $\pi$ Unleashed, which gives full bibliographic citations. 


EDIT: See also <http://mathworld.wolfram.com/PiFormulas.html> in particular formulas 93-96 where each term gives another 50 digits of $1/\pi$ (which I guess corresponds to $r$ roughly $10^{50}$). Somehow, the series for $1/\pi$ seem to do better than those for $\pi$. I know there are people who would like us to abandon $\pi$ in favor of $2\pi$, but maybe we should really be expressing things in terms of $1/(2\pi)$. 


