Sửa đề: \(A=\frac{2}{1\cdot5}+\frac{3}{5\cdot11}+\frac{4}{11\cdot19}+\frac{5}{19\cdot29}+\frac{6}{29\cdot41}\)
\(=\frac12\left(\frac{4}{1\cdot5}+\frac{6}{5\cdot11}+\frac{8}{11\cdot19}+\frac{10}{19\cdot29}+\frac{12}{29\cdot41}\right)\)
\(=\frac12\left(1-\frac15+\frac15-\frac{1}{11}+\frac{1}{11}-\frac{1}{19}+\frac{1}{19}-\frac{1}{29}+\frac{1}{29}-\frac{1}{41}\right)\)
\(=\frac12\left(1-\frac{1}{41}\right)=\frac12\cdot\frac{40}{41}=\frac{20}{41}\)
\(B=\frac{1}{1\cdot4}+\frac{2}{4\cdot10}+\frac{3}{10\cdot19}+\frac{4}{19\cdot31}\)
\(=\frac13\left(\frac{3}{1\cdot4}+\frac{6}{4\cdot10}+\frac{9}{10\cdot19}+\frac{12}{19\cdot31}\right)\)
\(=\frac13\left(1-\frac14+\frac14-\frac{1}{10}+\frac{1}{10}-\frac{1}{19}+\frac{1}{19}-\frac{1}{31}\right)=\frac13\left(1-\frac{1}{31}\right)=\frac13\cdot\frac{30}{31}=\frac{10}{31}\)
\(\frac{A}{B}=\frac{20}{41}:\frac{10}{31}=\frac{20}{41}\cdot\frac{31}{10}=\frac{2\cdot31}{41}=\frac{62}{41}\) >1
=>A>B
