# 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

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

2021-03-15 19:12:00.328586 UTC