Ta có :
\(\dfrac{1}{5^2}+\dfrac{1}{6^2}+\dfrac{1}{7^2}+..............+\dfrac{1}{100^2}< \dfrac{1}{4.5}+\dfrac{1}{5.6}+\dfrac{1}{7.8}+..................+\dfrac{1}{100.101}\)Đặt : \(A=\dfrac{1}{4.5}+\dfrac{1}{5.6}+\dfrac{1}{7.8}+..............+\dfrac{1}{100.101}\)
\(B=\dfrac{1}{5^2}+\dfrac{1}{6^2}+\dfrac{1}{7^2}+..................+\dfrac{1}{100^2}\)
\(A=\dfrac{1}{4.5}+\dfrac{1}{5.6}+\dfrac{1}{6.7}+............+\dfrac{1}{99.100}\)
\(A=\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+...................+\dfrac{1}{99}-\dfrac{1}{100}\)
\(A=\dfrac{1}{4}-\dfrac{1}{100}\)
\(A=\dfrac{6}{25}\)
Mà \(\dfrac{1}{6}< \dfrac{6}{25}< \dfrac{1}{4}\)
Ta lại có : \(A< \dfrac{6}{25}\)
Vậy \(\dfrac{1}{6}< \dfrac{1}{5^2}+\dfrac{1}{6^2}+\dfrac{1}{7^2}+............+\dfrac{1}{100^2}< \dfrac{1}{4}\)
~ Học tốt ~
