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 $

Ta có:

1/2.2 +1/3.3+1/4.4+....+1/100.100=1/2²+1/3²+1/4²+....+1/100²

Vì: 1/2²<1/1.2

     1/3²<1/2.3

     1/4²<1/3.4

     .....

     1/100²<1/99.100

→1/2²+1/3²+1/4²+....+1/100² < 1/1.2+1/2.3+1/3.4+....+1/99.100

→1/2²+1/3²+1/4²+....+1/100² < 1/1-1/2+1/2-1/3+1/3-1/4+....+1/99-1/100

→1-1/100 <1

→1/2²+1/3²+1/4²+....+1/100² < 1

→1/2.2 +1/3.3+1/4.4+....+1/100.100

Chúc bạn học tốt, nhớ vote+cảm ơn+ctlhn cho mik nhé!