ta có :\(\frac{1}{5^2}<\frac{1}{4.5}\)
\(\frac{1}{6^2}<\frac{1}{5.6}\)
\(\frac{1}{7^2}<\frac{1}{6.7}\)
.....
\(\frac{1}{100^2}<\frac{1}{99.100}\)
\(\Rightarrow A<\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}+...+\frac{1}{99.100}\)
\(=\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}+...+\frac{1}{99}-\frac{1}{100}=\frac{1}{4}-\frac{1}{100}<\frac{1}{4}\) (1)
Ta có : \(\frac{1}{5.6}<\frac{1}{5^2}\)'
\(\frac{1}{6.7}<\frac{1}{6^2}\)
....\(\frac{1}{100.101}<\frac{1}{100^2}\)
\(\frac{1}{5.6}+\frac{1}{6.7}+\frac{1}{7.8}+...+\frac{1}{100.101}\) <A
\(=\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+....+\frac{1}{100}-\frac{1}{101}\) <A
\(\frac{1}{5}-\frac{1}{101}\) <A
mà \(\frac{96}{5.101}=\frac{96}{505}>\frac{96}{576}\)
hay \(A>\frac{1}{6}\) (2)
từ (1); và (2) suy ra \(\frac{1}{6}<\frac{1}{5^2}+\frac{1}{6^2}+\frac{1}{7^2}+..+\frac{1}{100^2}<\frac{1}{4}\) (đpcm)
đây là cách dễ hiểu nhất nhé