\(A=\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2014}}{\dfrac{2013}{1}+\dfrac{2012}{2}+...+\dfrac{1}{2013}}\)
\(=\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2014}}{\left(\dfrac{2012}{2}+1\right)+\left(\dfrac{2011}{3}+1\right)+...+\left(\dfrac{1}{2013}+1\right)+\dfrac{2014}{2014}}\)
\(=\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2014}}{2014\left(\dfrac{1}{2}+\dfrac{1}{.3}+...+\dfrac{1}{2014}\right)}\)
\(=\dfrac{1}{2014}\)