Question

So sánh A= 1009/2020 và B=(1-1/2²)(1-1/3²)(1-1/4²)...(1-1/100²)

Answer 1

Ta có: \(B=\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\cdot\ldots\cdot\left(1-\frac{1}{100^2}\right)\)

\(=\left(1-\frac12\right)\left(1-\frac13\right)\cdot\ldots\cdot\left(1-\frac{1}{100}\right)\left(1+\frac12\right)\left(1+\frac13\right)\cdot\ldots\cdot\left(1+\frac{1}{100}\right)\)

\(=\frac12\cdot\frac23\cdot\ldots\cdot\frac{99}{100}\cdot\frac32\cdot\frac43\cdot\ldots\cdot\frac{101}{100}\)

\(=\frac{1}{100}\cdot\frac{101}{2}=\frac{101}{200}>\frac{100}{200}=\frac12\)

Ta có: \(\frac{1009}{2020}<\frac{1010}{2020}\)

=>\(A<\frac12\)

=>\(A<\frac12

=>A<B