Ta có :
$\dfrac{1}{2.2}< \dfrac{1}{1.2}$
$\dfrac{1}{3.3} < \dfrac{1}{2.3}$
$\dfrac{1}{4.4} < \dfrac{1}{3.4}$
$...................$
$\dfrac{1}{100.100} < \dfrac{1}{99.100}$
Do đó :
$\dfrac{1}{2.2}+\dfrac{1}{3.3}+\dfrac{1}{4.4}+....+\dfrac{1}{100.100}$
$< \dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{99.100}$
$ = \dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{99}-\dfrac{1}{100} $
$ = 1-\dfrac{1}{100} < 1 $
Vậy : $\dfrac{1}{2.2}+\dfrac{1}{3.3}+\dfrac{1}{4.4}+....+\dfrac{1}{100.100} < 1 $
