MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 (, 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?


share|cite|improve this question
I'm not sure I get your question. Why look at the radius of convergence? – Thierry Zell Feb 22 '11 at 22:00
@Thierry: a large ROC means that $a_n\to0$ rapidly, so that the partial sums are good approximations. A more fundamental phrasing of the question would be to ask for the function $\alpha(n)$ that can be computed with the fewest flops (asymptotically) and which satisfies $|\pi-\alpha(n)|<1/n$. – Kevin O'Bryant Feb 22 '11 at 22:11
But if $\sum a_nz^n$ has radius of convergence $r$, then $\sum b_nz^n$, where $b_n=a_n/2^n$, has radius of convergence $2r$, and doesn't do a better job of getting $\pi$. Are we assuming that $\pi$ is necessarily $f(1)$? – Gerry Myerson Feb 22 '11 at 22:15
Depends what you mean by best? If you want to find a specific digit, then something like the BBP formula should suit your needs. Otherwise the Chudnovsky modification of Ramanujan's formula is very fast. – Alex R. Feb 22 '11 at 22:30
The question should be rephrased. Instead of "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", I suggest: "Among all power series $f(x)=\sum_{n=0}^\infty a_n x^n$ with $f(1)=\pi$, and with the extra property that $a_{n+1}/a_n$ is a rational expression of $n$, which one has the largest radius of convergence?" – André Henriques Feb 22 '11 at 23:19
up vote 14 down vote accepted

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 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 $1/(2\pi)$.

share|cite|improve this answer

In Almkvist-Krattenthaler-Petersson:"Some new formulas for Pi" arXiv 2003?, Exp. Math 12(2003) 441-456 it is shown that there exist formulas for Pi where each term gives N new digits for any given N. Gert Almkvist

share|cite|improve this answer
@Gert, welcome to MO! – Gerry Myerson Apr 2 '11 at 22:45

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.