Question

chứng tỏ rằng B=1/2²+1/3²+1/4²+1/5²+1/6²+1/7²+1/8² <1

Answer 1

Đặt \(A=\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+...+\dfrac{1}{7\cdot8}\)

Dễ thấy: \(B=\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{8^2}\)\(< A=\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+...+\dfrac{1}{7\cdot8}\left(1\right)\)

Ta có:\(A=\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+...+\dfrac{1}{7\cdot8}\)

\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{7}-\dfrac{1}{8}\)

\(=1-\dfrac{1}{8}< 1\left(2\right)\)

Từ \((1);(2)\) ta có: \(B< A< 1\Rightarrow B< 1\)