Question

Chứng minh rằng: \(\frac{3}{1^2.2^2}\)+ \(\frac{5}{2^2.3^2}\)+ \(\frac{7}{3^2.4^2}\)+ ... + \(\frac{19}{9^2.10^2}\)<1

Answer 1

Đặt BT là A

\(\Rightarrow A=\frac{2^2-1^2}{1^2.2^2}+\frac{3^2-2^2}{2^2.3^2}+....+\frac{10^2-9^2}{9^2.10^2}\)

\(A=\frac{1}{1^2}-\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{2^2}+....+\frac{1}{9^2}-\frac{1}{10^2}\)

\(A=1-\frac{1}{10^2}< 1\)

=> A<1(đpcm)