Which series converging to pi is the fastest?

Question

Of course pi = 4 / (1-1/3+1/5-1/7+1/9-...)

and there are other infinite series that converge to pi.


But is this the fastest converging series?

Which series, defined using only integers and converging to pi is the fastest?

If I wanted to approximate pi using the first N terms of some series, what series would always get me the closest approximation for any given N? (is this known and is it proven?)

thanks.

Answer 1

Short answer: Chudnovsky formula.

There are many better than the basic arctan(pi/4) you mention or its arctangent-based infinite series summations cousins, the Machin-like formulae
(e.g. pi/4 = 4arctan(1/5) - arctan(1/239) and others)
or the Euler formula all converge faster.

(The Archimedes (upper- and lower-bounds fractions) method sucks totally for convergence.)

The Chudnovsky algorithm (1987) builds on Ramanujan's formulae (1910s).
It uses a hypergeometric series (=> very fast, and precision increases exponentially with each step).
1/pi =12 Sum (-1)^k (6k)!(13591409 + 545140134k)/(3k)!(k!)³ (640320)^((3k+3)/2)

which the Chudnovsky brothers used to achieve the then-world record of two billion digits in the early 1990s on a home-built supercomputer.

The original Ramanujan Formula converges very fast and uses only integers (and sqrt(2) on the LHS):
1/ 2pi sqrt(2) = 1103/99^2 + 27493/99^6 * (1/2)(1*3/4²) + ...

There may well be faster methods derived since but I would say Chudnovksy is "good-enough".

PS When you say "fastest-converging" it differs whether you mean "with the fewest iterations" (which you probably do) or "with the minimal computational effort".
Since obviously each iteration of Chudnovsky packs a lot of calculation (factorials, exponentiation), you would say it is "fast". Yet the arctangent-based formulae typically only involve a simple integer division.
(It would be interesting to graph the precision/computational-effort graph of where Chudnovsky passes out each of the other methods.)

See the bottom link, looks like a very readable book:
"The Number Pi" - by Pierre Eymard, Jean Pierre Lafon

Answer 2

Like every one above said, since it's periodic, why use large values. Of course, the usual Taylor expansion uses arguments in radians, not degrees. The only reason that I can see for an argument of 100 is if it were degrees. You do need to convert to radians first and reduce your argument to 0-2*pi. Even then the expansion can take a while time to converg. For an argument of 2*pi it takes 10 terms to get to 1%. You can avoind this 2 ways. The easy way is to convert your argument to the range of 0-pi/2 and apply the appropriate zero shifts and sign changes. You can also construct your own expansion. The general form of a Taytlor expansion is: SumOver_n(f(n)(a) * (x-a)^n / n!) Where f(n)(a) is the nth derivative of the function evaluated at 'a'. The usual cosine expansion is expanded around zero (meaning a=0) and therefore converges quickly near zero) you can construct a custom expansion about any point, just by selecting a value of 'a' that is near your point of interest.

Answer 3

The fastest converging formulas that aren't terrribly complicated utilize arctan identities. For example, in 1706 Machin developed a good one that goes:

π/4 = 4 * arctan(1/5) - arctan(1/239)

The calculation of π using this formula is accomplished by using the Taylor series for arctan(x), which is:

arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ...

You'll note that, since Machin's pi formula uses the numbers 1/5 and 1/239, the terms in the Taylor series get small very quickly: arctan(1/5) = 1/5 - 1/375 + 1/15625 - ... That allows you to compute pi to several decimal places with only a handful of calculations. Machin himself calculated pi to 100 decimal places using his formula, pretty impressive when you consider this was done before computers.

* * * *

Hunting around the Internet, here's a good and very readable Web page that gives everything I just said in more detail:

http://milan.milanovic.org/math/english/pi/machin.html

Answer 4

Chudnovsky Formula

Answer 5

Juan, you are a very intelligent person. But if you took 10% of your thoughts that you put into theothretical math and put it towards social applications such as helping the difficulties surrounding the distribution of wealth in society to greater incentives to politicians to lessen influence by special interest (some might call this corruption), your intelligence could have greater social value. Good luck with pi. It (pi) will always be there (infinity), the people around you won't (50years).

Answer 6

http://mathworld.wolfram.com/PiFormulas.html<--go to this site..