Fungrim entry: 042551

{e}^{\pi} = 32 \prod_{n=0}^{\infty} {\left(\frac{a_{n + 1}}{a_{n}}\right)}^{{2}^{1 - n}}\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(1, \frac{1}{\sqrt{2}}\right)

References:

https://doi.org/10.2307/2005327

TeX:

{e}^{\pi} = 32 \prod_{n=0}^{\infty} {\left(\frac{a_{n + 1}}{a_{n}}\right)}^{{2}^{1 - n}}\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(1, \frac{1}{\sqrt{2}}\right)

Definitions:

Fungrim symbol	Notation	Short description
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`Product`	$\prod_{n} f(n)$	Product
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`AGMSequence`	$\operatorname{agm}_{n}\!\left(a, b\right)$	Convergents in AGM iteration
`Sqrt`	$\sqrt{z}$	Principal square root

Source code for this entry:

Entry(ID("042551"),
    Formula(Equal(Exp(Pi), Where(Mul(32, Product(Pow(Div(a_(Add(n, 1)), a_(n)), Pow(2, Sub(1, n))), For(n, 0, Infinity))), Def(Tuple(a_(n), b_(n)), AGMSequence(n, 1, Div(1, Sqrt(2))))))),
    References("https://doi.org/10.2307/2005327"))

Topics using this entry

Arithmetic-geometric mean