Proving sum of infinite series My statistics textbook states that $\sum\limits_{n=0}^{\infty} \begin{pmatrix}2n \\ n \end{pmatrix}(pqs^2)^n=\frac{1}{\sqrt{1-4pqs^2}}$ 
where $|s|<1$, $p \in ]0,1[$ and $q=1-p$.
That thing is that prof of this is omitted since its "not relevant". How would one prove it? 
Soo far i have never calculated a sum involving a binomial coefficient. 

It is a consequence of the generalized binomial theorem, for $|x|<1$,
$$(1+x)^{-1/2}=\sum_{n=0}^{\infty}\binom{-1/2}{n}x^n,$$
where
$$\binom{-1/2}{n}=\frac{(-1/2)(-1/2-1)\cdots(-1/2-(n-1))}{n!}=\frac{(-1)^n}{4^n}\binom{2n}{n}.$$
Hence for $|4pqs^2|<1$,
$$\frac{1}{\sqrt{1-4pqs^2}}=(1+(-4pqs^2))^{-1/2}=\sum_{n=0}^{\infty}\binom{-1/2}{n}(-4pqs^2)^n\\=\sum_{n=0}^{\infty}\frac{(-1)^n}{4^n}\binom{2n}{n}(-4pqs^2)^n=\sum_{n=0}^{\infty}\binom{2n}{n}(pqs^2)^n.$$
Note that if $p\in(0,1)$ and $q=1-p$ then $pq\leq 1/4$ and for $|s|<1$ the condition $|4pqs^2|<1$ is satisfied.
