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Fungrim entry: 13c539

π(an+bn)21j=0n2jcj22n+8eπ2n+1   where (an,bn)=agmn ⁣(1,12),  cn=anbn\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
Absz\left|z\right| Absolute value
Piπ\pi The constant pi (3.14...)
Powab{a}^{b} Power
Sumnf(n)\sum_{n} f(n) Sum
Expez{e}^{z} Exponential function
AGMSequenceagmn ⁣(a,b)\operatorname{agm}_{n}\!\left(a, b\right) Convergents in AGM iteration
Sqrtz\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"))

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2021-03-15 19:12:00.328586 UTC