(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:
$$\begin{align}\pi
&= 3+\frac{1}{8}+\frac{9}{640}+\frac{15}{7168}+\frac{35}{98304}+\frac{189}{2883584}+\frac{693}{54525952}+\frac{429}{167772160} + \dots\\
&= 3\left(1+\frac{1}{24}+\frac{1}{24}\frac{9}{80}+\frac{1}{24}\frac{9}{80}\frac{25}{168}+\frac{1}{24}\frac{9}{80}\frac{25}{168}\frac{49}{288}+\frac{1}{24}\frac{9}{80}\frac{25}{168}\frac{49}{288}\frac{81}{440}+\frac{1}{24}\frac{9}{80}\frac{25}{168}\frac{49}{288}\frac{81}{440}\frac{121}{624}+\frac{1}{24}\frac{9}{80}\frac{25}{168}\frac{49}{288}\frac{81}{440}\frac{121}{624}\frac{169}{840}+\dots\right)
\end{align}$$
or, more compactly,
$$\pi = 3\left(1 + \sum_{n=1}^{\infty}\prod_{k=1}^{n}\frac{(2k-1)^2}{8k(2k+1)}\right)$$
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.
Best Answer
That is the Taylor series of $\arcsin(x)$ at $x=1/2$ (times 6).