$S=\frac{1}{2!2012!}+\frac{1}{4!2010!}+\frac{1}{6!2008!}+...+\frac{1}{2012!2!}+\frac{1}{2014!0!}$
$=\frac{1}{2014!}(\frac{2014!}{2!2012!}+\frac{2014!}{4!2010!}+\frac{2014!}{6!2008!}+...+\frac{2014!}{2012!2!}+\frac{2014!}{2014!0!})$
$=\frac{1}{2014!}(C^{2}_{2014}+C^{4}_{2014}+C^{6}_{2014}+...+C^{2012}_{2014}+C^{2014}_{2014})$
$=\frac{1}{2014!}[(C^{0}_{2014}+C^{2}_{2014}+C^{4}_{2014}+C^{6}_{2014}+...+C^{2012}_{2014}+C^{2014}_{2014})-1]$
$=\frac{1}{2014!}(2^{2013}-1)$
$=\frac{2^{2013}-1}{2014!}$.