# Fungrim entry: 6d9ceb

$\pi = \frac{4 {\left(\operatorname{agm}\!\left(1, \frac{1}{\sqrt{2}}\right)\right)}^{2}}{1 - \sum_{j=0}^{\infty} {2}^{j} c_{j}^{2}} = \lim_{n \to \infty} \frac{{\left(a_{n} + b_{n}\right)}^{2}}{1 - \sum_{j=0}^{n} {2}^{j} c_{j}^{2}}\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(1, \frac{1}{\sqrt{2}}\right),\;c_{n} = a_{n} - b_{n}$
TeX:
\pi = \frac{4 {\left(\operatorname{agm}\!\left(1, \frac{1}{\sqrt{2}}\right)\right)}^{2}}{1 - \sum_{j=0}^{\infty} {2}^{j} c_{j}^{2}} = \lim_{n \to \infty} \frac{{\left(a_{n} + b_{n}\right)}^{2}}{1 - \sum_{j=0}^{n} {2}^{j} c_{j}^{2}}\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(1, \frac{1}{\sqrt{2}}\right),\;c_{n} = a_{n} - b_{n}
Definitions:
Fungrim symbol Notation Short description
Pi$\pi$ The constant pi (3.14...)
Pow${a}^{b}$ Power
AGM$\operatorname{agm}\!\left(a, b\right)$ Arithmetic-geometric mean
Sqrt$\sqrt{z}$ Principal square root
Sum$\sum_{n} f(n)$ Sum
Infinity$\infty$ Positive infinity
SequenceLimit$\lim_{n \to a} f(n)$ Limiting value of sequence
AGMSequence$\operatorname{agm}_{n}\!\left(a, b\right)$ Convergents in AGM iteration
Source code for this entry:
Entry(ID("6d9ceb"),
Formula(Where(Equal(Pi, Div(Mul(4, Pow(AGM(1, Div(1, Sqrt(2))), 2)), Sub(1, Sum(Mul(Pow(2, j), Pow(c_(j), 2)), For(j, 0, Infinity)))), SequenceLimit(Div(Pow(Add(a_(n), b_(n)), 2), Sub(1, Sum(Mul(Pow(2, j), Pow(c_(j), 2)), For(j, 0, n)))), For(n, Infinity))), Def(Tuple(a_(n), b_(n)), AGMSequence(n, 1, Div(1, Sqrt(2)))), Def(c_(n), Sub(a_(n), b_(n))))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC