Fungrim entry: cfefa9

\operatorname{agm}\!\left(1, 1 + x\right) = \sum_{n=0}^{\infty} \frac{\text{A060691}\!\left(n\right)}{{8}^{n}} {\left(-x\right)}^{n}

Assumptions:

x \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
`AGM`	$\operatorname{agm}\!\left(a, b\right)$	Arithmetic-geometric mean
`Sum`	$\sum_{n} f(n)$	Sum
`SloaneA`	$\text{A00000X}\!\left(n\right)$	Sequence X in Sloane's OEIS
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	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))))

Topics using this entry

Arithmetic-geometric mean