Question

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Answer 1

\(A=\left(\dfrac{1}{2^2}-1\right)\left(\dfrac{1}{3^2}-1\right)\left(\dfrac{1}{4^2}-1\right)...\left(\dfrac{1}{2013^2}-1\right)\left(\dfrac{1}{2014^2}-1\right)\)

\(=-\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{3^2}\right)\left(1-\dfrac{1}{4^2}\right)...\left(1-\dfrac{1}{2013^2}\right)\left(1-\dfrac{1}{2014^2}\right)\)

\(=-\dfrac{1}{2}.\dfrac{3}{2}.\dfrac{2}{3}.\dfrac{4}{3}.\dfrac{3}{4}.\dfrac{5}{4}...\dfrac{2012}{2013}.\dfrac{2014}{2013}.\dfrac{2013}{2014}.\dfrac{2015}{2014}\)

\(=-\dfrac{1}{2}.\dfrac{2015}{2014}< -\dfrac{1}{2}=B\)

\(\Rightarrow A< B\)