# Fungrim entry: c9bcf7

$\sum_{n=0}^{\infty} \frac{1}{{2 n \choose n}} {x}^{n} = \,{}_2F_1\!\left(1, 1, \frac{1}{2}, \frac{x}{4}\right) = \frac{1}{1 - u} + \frac{\sqrt{u} \operatorname{asin}\!\left(\sqrt{u}\right)}{{\left(1 - u\right)}^{3 / 2}}\; \text{ where } u = \frac{x}{4}$
Assumptions:$x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < 4$
TeX:
\sum_{n=0}^{\infty} \frac{1}{{2 n \choose n}} {x}^{n} = \,{}_2F_1\!\left(1, 1, \frac{1}{2}, \frac{x}{4}\right) = \frac{1}{1 - u} + \frac{\sqrt{u} \operatorname{asin}\!\left(\sqrt{u}\right)}{{\left(1 - u\right)}^{3 / 2}}\; \text{ where } u = \frac{x}{4}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < 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
Hypergeometric2F1$\,{}_2F_1\!\left(a, b, c, z\right)$ Gauss hypergeometric function
Sqrt$\sqrt{z}$ Principal square root
CC$\mathbb{C}$ Complex numbers
Abs$\left|z\right|$ Absolute value
Source code for this entry:
Entry(ID("c9bcf7"),
Formula(Equal(Sum(Mul(Div(1, Binomial(Mul(2, n), n)), Pow(x, n)), For(n, 0, Infinity)), Hypergeometric2F1(1, 1, Div(1, 2), Div(x, 4)), Where(Add(Div(1, Sub(1, u)), Div(Mul(Sqrt(u), Asin(Sqrt(u))), Pow(Sub(1, u), Div(3, 2)))), Equal(u, Div(x, 4))))),
Variables(x),
Assumptions(And(Element(x, CC), Less(Abs(x), 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