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Fungrim entry: 6d9ceb

π=4(agm ⁣(1,12))21j=02jcj2=limn(an+bn)21j=0n2jcj2   where (an,bn)=agmn ⁣(1,12),  cn=anbn\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
Piπ\pi The constant pi (3.14...)
Powab{a}^{b} Power
AGMagm ⁣(a,b)\operatorname{agm}\!\left(a, b\right) Arithmetic-geometric mean
Sqrtz\sqrt{z} Principal square root
Sumnf(n)\sum_{n} f(n) Sum
Infinity\infty Positive infinity
SequenceLimitlimnaf(n)\lim_{n \to a} f(n) Limiting value of sequence
AGMSequenceagmn ⁣(a,b)\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))))))

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2020-08-27 09:56:25.682319 UTC