$\dfrac{1}{2^2}<\dfrac{1}{1.2}$
..........
$\dfrac{1}{100^2}<\dfrac{1}{99.100}$
$⇒F<\dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{99.100}$
$⇒F<\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{99}-\dfrac{1}{100}$
$⇒F<1-\dfrac{1}{100}=\dfrac{99}{100}$
$⇒F<\dfrac{99}{100}$ (1)
$\dfrac{1}{2^2}>\dfrac{1}{2.3}$
.........
$\dfrac{1}{100^2}>\dfrac{1}{100.101}$
$⇒F>\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{100.101}$
$⇒F>\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{100}-\dfrac{1}{101}$
$⇒F>\dfrac{1}{2}-\dfrac{1}{101}=\dfrac{99}{202}$
$⇒F>\dfrac{99}{202}$ (2)
Từ (1), (2) $⇒\dfrac{99}{202}<F<\dfrac{99}{100}$
