\(B=\frac{1}{2^2}+\frac{1}{3^2}+....+\frac{1}{8^2}< \frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{7.8}=1-\frac{1}{8}< 1\)
\(B=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{8^2}\)
vì \(\frac{1}{2^2}>\frac{1}{1\cdot2}\)
\(\frac{1}{3^2}< \frac{1}{2\cdot3}\)
\(\frac{1}{4^2}< \frac{1}{3\cdot4}\)
\(...\)
\(\frac{1}{8^2}< \frac{1}{7\cdot8}\)
nên \(A< \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{7\cdot8}\) (1)
\(B=\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{7\cdot8}\)
\(B=\frac{2-1}{1\cdot2}+\frac{3-2}{2\cdot3}+\frac{4-3}{3\cdot4}+...+\frac{8-7}{7\cdot8}\)
\(B=\left(\frac{2}{1\cdot2}-\frac{1}{1\cdot2}\right)+\left(\frac{3}{2\cdot3}-\frac{2}{2\cdot3}\right)+...+\left(\frac{8}{7\cdot8}-\frac{1}{7\cdot8}\right)\)
\(B=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{7}-\frac{1}{8}\)
\(B=1-\frac{1}{8}\)
\(B=\frac{7}{8}< 1\) (2)
(1)(2) \(\Rightarrow A< B< 1\)
\(\Rightarrow A< 1\) (đpct)