Conjecture: $\sum\limits_{n\geq0}\left(\frac12\right)^n\prod\limits_{k=1}^{n}\frac{2n-2k+1}{2n-2k+2}=\sqrt2$ I am trying to solve a problem I made form myself: proving that 
$$\sum_{n\geq0}\left(\frac12\right)^n\prod_{k=1}^{n}\frac{2n-2k+1}{2n-2k+2}=\sqrt2$$
The highly accurate powers of Desmos seem to confirm my hunch. But how do I prove it? 
I was just messing around with products and generating functions, and then I noticed that the numerical value of the sum in question was suspiciously similar to $\sqrt2$, so I conjectured the result. Unfortunately I found this completely by accident and have absolutely no idea of how to prove it. 
Feeble attempt:
Define $$S(x)=\sum_{n\geq0}x^n\prod_{k=1}^{n}\frac{2n-2k+1}{2n-2k+2}$$
Which Wolfram says is equal to $$S(x)=\sum_{n\geq0}x^n\frac{(1/2-n)_n}{(-n)_n}$$
With $\displaystyle (x)_n=\frac{\Gamma(x+n)}{\Gamma(x)}$. But that doesn't really make sense because $\Gamma(0)$ is undefined. So all in all I'm just confused. 
Could I have some help?

Edit:
According to the comments, it suffices to prove that 
$$\sum_{n\geq1}\left(\frac12\right)^n\prod_{k=1}^{n}\frac{2n-2k+1}{2n-2k+2}=\sqrt2-1$$
 A: Another way is
$$
\eqalign{
  & \prod\limits_{k = 1}^n {{{2n - 2k + 1} \over {2n - 2k + 2}}}  = \prod\limits_{k = 1}^n {{{\left( {n - k} \right) + 1/2} \over {\left( {n - k} \right) + 1}}}
  = \prod\limits_{j = 0}^{n - 1} {{{j + 1/2} \over {j + 1}}}  =   \cr 
  &  = {{\left( {1/2} \right)^{\,\overline {\,n\,} } } \over {n!}}
 = {{\Gamma \left( {n + 1/2} \right)} \over {\Gamma \left( {1/2} \right)\Gamma \left( {n + 1} \right)}} =
  \left( \matrix{ n - 1/2 \cr   n \cr}  \right)
 = \left( { - 1} \right)^{\,n} \left( \matrix{   - 1/2 \cr   n \cr}  \right) \cr} 
$$
and then apply the binomial expansion.
A: The products can be rewritten as 
\begin{eqnarray*}
\sum_{n=0}^{\infty} \frac{(2n-1)!!}{(2n)!!} \frac{1}{2^n} = \sum_{n=0}^{\infty} \binom{2n}{n} \frac{1}{8^n}
\end{eqnarray*}
Now use
\begin{eqnarray*}
\sum_{n=0}^{\infty}  \binom{2n}{n} x^n = \frac{1}{\sqrt{1-4x}}.
\end{eqnarray*}
Edit:
\begin{eqnarray*}
\prod_{k=1}^{n}\frac{2n-2k+1}{2n-2k+2}&=&\prod_{k=1}^{n}\frac{2k-1}{2k} =\frac{(2n-1)!!}{(2n)!!}\\
&=&\frac{(2n)!}{(2^n n!)^2}=\binom{2n}{n} \frac{1}{2^{2n}}.
\end{eqnarray*}
