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\(A=\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{2012^2}+\dfrac{1}{2013^2}\)
\(A< \dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{2012.2013}+\dfrac{1}{2013.2014}\)
\(A< \dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{2012}-\dfrac{1}{2013}+\dfrac{1}{2013}-\dfrac{1}{2014}\)
\(A< \dfrac{1}{2}-\dfrac{1}{2014}\)
\(A< \dfrac{1007}{2014}-\dfrac{1}{2014}\)
\(A< \dfrac{1006}{2014}< 1\)
\(\Rightarrow A< 1\)
A= 1+(\(\dfrac{1}{2014}\)+1)+(\(\dfrac{2}{2013}\)+1)+...+(\(\dfrac{2013}{2}\)+1)
= \(\dfrac{2015}{2015}\)+(\(\dfrac{1}{2014}\)+1)+(\(\dfrac{2}{2013}\)+1)+...+(\(\dfrac{2013}{2}\)+1)
= 2015.(\(\dfrac{1}{2015}\)+\(\dfrac{1}{2014}\)+\(\dfrac{1}{2013}\)+...+\(\dfrac{1}{2}\))=2015.B
\(\Rightarrow\) \(\dfrac{A}{B}\)=2015
