# Fungrim entry: 2b2066

$\sum_{n=0}^{\infty} {2 n \choose n} {x}^{n} = \frac{1}{\sqrt{1 - 4 x}}$
Assumptions:$x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < \frac{1}{4}$
TeX:
\sum_{n=0}^{\infty} {2 n \choose n} {x}^{n} = \frac{1}{\sqrt{1 - 4 x}}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < \frac{1}{4}
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
Sqrt$\sqrt{z}$ Principal square root
CC$\mathbb{C}$ Complex numbers
Abs$\left|z\right|$ Absolute value
Source code for this entry:
Entry(ID("2b2066"),
Formula(Equal(Sum(Mul(Binomial(Mul(2, n), n), Pow(x, n)), For(n, 0, Infinity)), Div(1, Sqrt(Sub(1, Mul(4, x)))))),
Variables(x),
Assumptions(And(Element(x, CC), Less(Abs(x), Div(1, 4)))))

## 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