How can I prove that $\frac{1}{\sqrt{1-4x}}=\sum_{n=0}^{\infty} \binom {2n} {n} x^n$ How can I prove that $\frac{1}{\sqrt{1-4x}}=\sum_{n=0}^{\infty} \binom {2n} {n} x^n$? 
I know the I should use Newton's generalized binomial theorem which says that,
$$(1+z)^a=\sum_{n=0}^{\infty} \binom {a} {n}z^n=1+\binom{a}{1}z+\binom{a}{2}z^2+...$$
But how can I use it in this special case?

\begin{eqnarray*}
\binom{-1/2}{n} (-4)^n = \frac{(-1/2)(-3/2) \cdots (-1/2-(n-1))}{n!} (-4)^n=\frac{(-1)^n(2n-1)!!}{2^n n!} \frac{(2n)!!}{2^n n!} (-4)^n = \binom{2n}{n}.
\end{eqnarray*}
