Ta có: `1/2^2 < 1/1.2 ; 1/3^2 < 1/2.3; ..... ; 1/9^2 < 1/8.9`
`=> 1/2^2 + 1/3^2 + 1/4^2+.........+ 1/9^2 < 1/1.2 + 1/2.3 + .... + 1/8.9`
`=> A < 1/1 -1/2 + 1/2 -1/3 +....+ 1/8- 1/9`
`=>A < 1/1 -1/9`
`=> A < 8/9` `(1)`
Lại có: `1/2^2 > 1/2.3 + 1/3^2 > 1/3.4 ; .........; 1/9^2 > 1/9.10`
`=> 1/2^2 + 1/3^2 + 1/4^2+.........+ 1/9^2 > 1/2.3 + 1/3.4 +...+ 1/9.10`
`=> A > 1/2 -1/3 + 1/3 -1/4 +....+ 1/9-1/10`
`=> A > 1/2 -1/10 = 2/5` `(2)`
Từ `(1)` và `(2)` `=> 8/9 > A > 2/5`
Vậy `8/9 > A > 2/5`
