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Fungrim entry: 7c014b

k=0(zk)xk=(1+x)z\sum_{k=0}^{\infty} {z \choose k} {x}^{k} = {\left(1 + x\right)}^{z}
Assumptions:zCandxCandx<1z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|x\right| < 1
Alternative assumptions:zZ0andzCz \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
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Binomial(nk){n \choose k} Binomial coefficient
Powab{a}^{b} Power
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
Absz\left|z\right| Absolute value
ZZGreaterEqualZn\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)), Tuple(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))))

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

2019-08-21 11:44:15.926409 UTC