Fairly Impossible#1

Using the fact that, $\sum_{n=1}^{\infty}\frac{\Gamma(x+n-1)\Gamma(y+1)}{\Gamma(n+x+y)}$ $=$ $\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ .Then simply letting $y=1/2$ and $x=1/2$ ,we have our sum with the result as $\pi$ ,since $\Gamma(1/2)$ $=$ $\sqrt\pi$ .Then,

$\frac{\pi}{\pi}$ $=$ $1$ ,for the proof of this result look at the discussion on the iterated properties of Beta Function.