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Fungrim entry: cfefa9

agm ⁣(1,1+x)=n=0A060691 ⁣(n)8n(x)n\operatorname{agm}\!\left(1, 1 + x\right) = \sum_{n=0}^{\infty} \frac{\text{A060691}\!\left(n\right)}{{8}^{n}} {\left(-x\right)}^{n}
Assumptions:xC  and  x<1x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < 1
References:
  • Sequence A060691 in Sloane's On-Line Encyclopedia of Integer Sequences (OEIS)
TeX:
\operatorname{agm}\!\left(1, 1 + x\right) = \sum_{n=0}^{\infty} \frac{\text{A060691}\!\left(n\right)}{{8}^{n}} {\left(-x\right)}^{n}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < 1
Definitions:
Fungrim symbol Notation Short description
AGMagm ⁣(a,b)\operatorname{agm}\!\left(a, b\right) Arithmetic-geometric mean
Sumnf(n)\sum_{n} f(n) Sum
SloaneAA00000X ⁣(n)\text{A00000X}\!\left(n\right) Sequence X in Sloane's OEIS
Powab{a}^{b} Power
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
Absz\left|z\right| Absolute value
Source code for this entry:
Entry(ID("cfefa9"),
    Formula(Equal(AGM(1, Add(1, x)), Sum(Mul(Div(SloaneA("060691", n), Pow(8, n)), Pow(Neg(x), n)), For(n, 0, Infinity)))),
    Variables(x),
    Assumptions(And(Element(x, CC), Less(Abs(x), 1))))

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

2020-08-27 09:56:25.682319 UTC