The generating function of central binomial coefficients

1. Coefficients. For $\alpha=-\frac12$,
$$\binom{-1/2}n=\frac{\left(-\frac12\right)\left(-\frac32\right)\cdots\left(-\frac{2n-1}2\right)}{n!}=\frac{(-1)^n(2n-1)!!}{2^nn!},$$
where $(2n-1)!!=1\cdot3\cdot5\cdots(2n-1)=\frac{(2n)!}{2^nn!}$. Hence
$$\binom{-1/2}n=\frac{(-1)^n(2n)!}{4^n(n!)^2}=\frac{(-1)^n}{4^n}\binom{2n}n.$$
With $u=-4x$, $\binom{-1/2}nu^n=\binom{2n}nx^n$, which proves the formula for $|4x|<1$.

2. Radius. The binomial series has radius $1$ in $u$, so $R=\frac14$ in $x$. This agrees with $\frac{\binom{2n+2}{n+1}}{\binom{2n}n}\to4$ and the root test result $\binom{2n}n^{1/n}\to4$.

Value. At $x=\frac15<\frac14$:
$$\sum_{n\ge0}\binom{2n}n\frac1{5^n}=\frac1{\sqrt{1-\frac45}}=\sqrt5.$$