Infinite series involving binomial and geometric How to find the following sum:
$$\sum_{x=0}^{\infty}\left(-\frac{1}{16}\right)^x \binom{2x}{x}$$

\begin{eqnarray*}
\binom{2n}{n} = \binom{-1/2}{n}(-4)^{n}.
\end{eqnarray*}
\begin{eqnarray*}
\sum_{n=0}^{\infty} \binom{2n}{n} \left( \frac{-1}{16} \right)^n = \sum_{n=0}^{\infty} \binom{-1/2}{n} \left( \frac{1}{4} \right)^n =\frac{1}{\sqrt{1+\frac{1}{4}}} =\color{red}{\frac{2}{\sqrt{5}}}.
\end{eqnarray*}
