# 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

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2021-03-15 19:12:00.328586 UTC