Cho: A = 1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + ... + 1/50^2. Chứng minh A < 2
1/2^2<1/(1.2)=1-1/2
1/3^2<1/(2.3)=1/2-1/3
.........
1/50^2<1/(49.50)=1/49-1/50
=>1/2^2 + 1/3^2 + 1/4^2 + ... + 1/50^2<1-1/2+1/2-1/3+.....+1/49-1/50=1-1/50<1
=> A = 1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + ... + 1/50^2<1+1=2
vậy A<2

ta có 1/2^2 < 1/1.2 ;1/3^2< 1/2.3 tương tự .....1/50^2 < 1/49.50
cộng từng vế BĐT ta được :
1/2^2+1/3^2 +....+1/50^2 < 1/1.2+1/2.3+...+1/49.50 =1-1/50=49/50
=> 1/1^2+1/2^2+1/3^2+....+1/50^2 < 1+49/50<2
=>đpcm