Question

Chứng minh \(\dfrac{1}{2}\) < (1-\(\dfrac{1}{2^2}\))(1-\(\dfrac{1}{3^2}\))(1-\(\dfrac{1}{4^2}\))....(1-\(\dfrac{1}{n^2}\))<1

( Với mọi n\(\in\)N và n \(\ge\) 2)

Answer 1

Ta có: \(1-\dfrac{1}{n^2}=\dfrac{\left(n-1\right)\left(n+1\right)}{n^2}\)

Thế vô bài toán ta được

\(\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{3^2}\right)...\left(1-\dfrac{1}{n^2}\right)=\dfrac{1.3}{2.2}.\dfrac{2.4}{3.3}...\dfrac{\left(n-1\right)\left(n+1\right)}{n.n}=\dfrac{1}{2}.\dfrac{n+1}{n}\)

Ta thấy

\(\dfrac{1}{2}.\dfrac{n}{n}< \dfrac{1}{2}.\dfrac{n+1}{n}< \dfrac{1}{2}.\dfrac{n+n}{n}\)

\(\Rightarrow\dfrac{1}{2}< \dfrac{1}{2}.\dfrac{n+1}{n}< 1\)

\(\Rightarrow\)ĐPCM