Question

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

và B = \(\frac{-1}{2}\). Hãy so sánh biểu thức A và B

nhanh nhé, mk cần gấp, ai đúng mk **** cho

Answer 1

Ta có \(-A=\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)...\left(1-\frac{1}{2014^2}\right)\)

\(=\left(\frac{2^2-1}{2^2}\right)\left(\frac{3^2-1}{3^2}\right)...\left(\frac{2014^2-1}{2014^2}\right)\)

\(=\frac{\left(2-1\right)\left(2+1\right)}{2^2}.\frac{\left(3-1\right)\left(3+1\right)}{3^2}...\frac{\left(2014-1\right)\left(2014+1\right)}{2014^2}\)

\(=\frac{1.3}{2.2}.\frac{2.4}{3.3}...\frac{2013.2015}{2014.2014}\)

\(=\frac{1.2...2013}{2.3...2014}.\frac{3.4...2015}{2.3...2014}\)

\(=\frac{1}{2014}.\frac{2015}{2}\)

\(=\frac{2015}{2014.2}>\frac{1}{2}\)hay -A>1/2

=>\(A< \frac{-1}{2}\)hay A<B