Cfrtjryk

(Don't be alarmed by the title; this is a question about mathematics, not programming.)

In the documentation for the decimal module in the Python Standard Library, an example is given for computing the digits of $pi$ to a given precision:

def pi():

    """Compute Pi to the current precision.



    >>> print(pi())

    3.141592653589793238462643383



    """

    getcontext().prec += 2  # extra digits for intermediate steps

    three = Decimal(3)      # substitute "three=3.0" for regular floats

    lasts, t, s, n, na, d, da = 0, three, 3, 1, 0, 0, 24

    while s != lasts:

        lasts = s

        n, na = n+na, na+8

        d, da = d+da, da+32

        t = (t * n) / d

        s += t

    getcontext().prec -= 2

    return +s               # unary plus applies the new precision

I was not able to find any reference for what formula or fact about $pi$ this computation uses, hence this question.

Translating from code into more typical mathematical notation, and using some calculation and observation, this amounts to a formula for $pi$ that begins like:

$$begin{align}pi
&= 3+frac{1}{8}+frac{9}{640}+frac{15}{7168}+frac{35}{98304}+frac{189}{2883584}+frac{693}{54525952}+frac{429}{167772160} + dots\
&= 3left(1+frac{1}{24}+frac{1}{24}frac{9}{80}+frac{1}{24}frac{9}{80}frac{25}{168}+frac{1}{24}frac{9}{80}frac{25}{168}frac{49}{288}+frac{1}{24}frac{9}{80}frac{25}{168}frac{49}{288}frac{81}{440}+frac{1}{24}frac{9}{80}frac{25}{168}frac{49}{288}frac{81}{440}frac{121}{624}+frac{1}{24}frac{9}{80}frac{25}{168}frac{49}{288}frac{81}{440}frac{121}{624}frac{169}{840}+dotsright)
end{align}$$

or, more compactly,

$$pi = 3left(1 + sum_{n=1}^{infty}prod_{k=1}^{n}frac{(2k-1)^2}{8k(2k+1)}right)$$

Is this a well-known formula for $pi$? How is it proved? How does it compare to other methods, in terms of how how quickly it converges, numerical stability issues, etc? At a glance I didn't see it on the Wikipedia page for List of formulae involving π or on the MathWorld page for Pi Formulas.

That is the Taylor series of $arcsin(x)$ at $x=1/2$ (times 6).

It just computing $pi = 6sin^{-1}left(frac12right)$ using the Taylor series expansion of
arcsine. For reference,

$$6sin^{-1}frac{t}{2} = 3t+frac{t^3}{8}+frac{9t^5}{640}+frac{15t^7}{7168}+frac{35 t^9}{98304} + cdots$$
and compare the coefficients with what you get.