Fungrim entry: 13c539

\left|\pi - \frac{{\left(a_{n} + b_{n}\right)}^{2}}{1 - \sum_{j=0}^{n} {2}^{j} c_{j}^{2}}\right| \le {2}^{n + 8} {e}^{-\pi {2}^{n + 1}}\; \text{ where } \left(a_{n}, b_{n}\right) = \operatorname{agm}_{n}\!\left(1, \frac{1}{\sqrt{2}}\right),\;c_{n} = a_{n} - b_{n}

References:

https://doi.org/10.2307/2005327

TeX:

\left|\pi - \frac{{\left(a_{n} + b_{n}\right)}^{2}}{1 - \sum_{j=0}^{n} {2}^{j} c_{j}^{2}}\right| \le {2}^{n + 8} {e}^{-\pi {2}^{n + 1}}\; \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
`Abs`	$\left\|z\right\|$	Absolute value
`Pi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`Sum`	$\sum_{n} f(n)$	Sum
`Exp`	${e}^{z}$	Exponential function
`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("13c539"),
    Formula(Where(LessEqual(Abs(Sub(Pi, Div(Pow(Add(a_(n), b_(n)), 2), Sub(1, Sum(Mul(Pow(2, j), Pow(c_(j), 2)), For(j, 0, n)))))), Mul(Pow(2, Add(n, 8)), Exp(Neg(Mul(Pi, Pow(2, Add(n, 1))))))), Def(Tuple(a_(n), b_(n)), AGMSequence(n, 1, Div(1, Sqrt(2)))), Def(c_(n), Sub(a_(n), b_(n))))),
    References("https://doi.org/10.2307/2005327"))

Fungrim entry: 13c539

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