$\text{ Đặt A=}$${\dfrac{1}{2^2} + \dfrac{1}{3^2} + \dfrac{1}{4^2}+...+\dfrac{1}{100^2}}$
$\text{ Ta thấy}$
${ \dfrac{1}{2^2}=\dfrac{1}{2.2}=\dfrac{1}{4}}$
${ \dfrac{1}{3^2}=\dfrac{1}{3.3} < \dfrac{1}{2.3}}$
${ \dfrac{1}{4^2}=\dfrac{1}{4.4} < \dfrac{1}{3.4}}$
....
${\dfrac{1}{100^2}=\dfrac{1}{100.100} < \dfrac{1}{99.100}}$
⇒ ${\dfrac{1}{2^2} + \dfrac{1}{3^2} + \dfrac{1}{4^2}+...+\dfrac{1}{100^2}}$ < ${\dfrac{1}{4}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{99.100}}$
⇒ ${A}$ < ${\dfrac{1}{4}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{99}+\dfrac{1}{100}}$
⇒ ${A}$ ${< \dfrac{1}{4}+\dfrac{1}{2}-\dfrac{1}{100}}$
⇒ ${A}$ ${< \dfrac{3}{4} - \dfrac{1}{100}}$
⇒ ${A}$${ < \dfrac{3}{4}}$
