1:
\(A=\dfrac{2}{3}\left(\dfrac{3}{60\cdot63}+\dfrac{3}{63\cdot66}+...+\dfrac{3}{117\cdot120}\right)+\dfrac{2}{2011}\)
\(=\dfrac{2}{3}\left(\dfrac{1}{60}-\dfrac{1}{63}+\dfrac{1}{63}-\dfrac{1}{66}+...+\dfrac{1}{117}-\dfrac{1}{120}\right)+\dfrac{2}{2011}\)
\(=\dfrac{2}{3}\cdot\dfrac{1}{120}+\dfrac{2}{2021}=\dfrac{2}{360}+\dfrac{2}{2021}\)
\(B=\dfrac{5}{4}\left(\dfrac{4}{40\cdot44}+\dfrac{4}{44\cdot48}+...+\dfrac{4}{76\cdot80}\right)+\dfrac{5}{2011}\)
\(=\dfrac{5}{4}\cdot\left(\dfrac{1}{40}-\dfrac{1}{44}+\dfrac{1}{44}-\dfrac{1}{48}+...+\dfrac{1}{76}-\dfrac{1}{80}\right)+\dfrac{5}{2011}\)
\(=\dfrac{5}{4}\cdot\dfrac{1}{80}+\dfrac{5}{2011}=\dfrac{5}{320}+\dfrac{5}{2011}\)
2/360<5/320
2/2011<5/2011
=>A<B
