\(A=\frac{1}{2!}+\frac{2}{3!}+...+\frac{2013}{2014!}=\frac{2-1}{2!}+\frac{3-1}{3!}+...+\frac{2014-1}{2014!}\)
\(=1-\frac{1}{2!}+\frac{3}{3!}-\frac{1}{3!}+...+\frac{2014}{2014!}-\frac{1}{2014!}\)
\(=1-\frac{1}{2!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{3!}-\frac{1}{4!}+...+\frac{1}{2013!}-\frac{1}{2014!}\)
\(=1-\frac{1}{2014!}