Question

so sánh A= (\(\dfrac{1}{2^2}\) -1) (\(\dfrac{1}{3^2}\) -1)..........(\(\dfrac{1}{100^2}\) -1) VỚI -\(\dfrac{1}{2}\)

Answer 1

\(\Rightarrow-A=\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{3^2}\right)...\left(1-\dfrac{1}{100^2}\right)\)

\(-A=\dfrac{3}{2^2}.\dfrac{8}{3^2}...\dfrac{9999}{100^2}\)

\(-A=\dfrac{1.3}{2.2}.\dfrac{2.4}{3.3}...\dfrac{99.101}{100.100}\)

\(-A=\dfrac{1.2...99}{2.3...100}.\dfrac{3.4...101}{2.3...100}\)

\(-A=\dfrac{1}{100}.\dfrac{101}{2}=\dfrac{101}{200}\)

\(\Rightarrow A=\dfrac{-101}{200}\)

Mà \(\dfrac{-101}{200}< \dfrac{-100}{200}=\dfrac{-1}{2}\)

\(\Rightarrow A< \dfrac{-1}{2}\)