Fungrim entry: 7c014b

\sum_{k=0}^{\infty} {z \choose k} {x}^{k} = {\left(1 + x\right)}^{z}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < 1

Alternative assumptions:

z \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

\sum_{k=0}^{\infty} {z \choose k} {x}^{k} = {\left(1 + x\right)}^{z}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < 1

z \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`Binomial`	${n \choose k}$	Binomial coefficient
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("7c014b"),
    Formula(Equal(Sum(Mul(Binomial(z, k), Pow(x, k)), For(k, 0, Infinity)), Pow(Add(1, x), z))),
    Variables(z, x),
    Assumptions(And(Element(z, CC), Element(x, CC), Less(Abs(x), 1)), And(Element(z, ZZGreaterEqual(0)), Element(z, CC))))

Topics using this entry

Factorials and binomial coefficients