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

Factorials and binomial coefficients