Ta có :
`1/2^2 < 1 - 1/2`
`1/3^2 < 1/2 - 1/3`
`1/4^2 < 1/3 - 1/4`
`...`
`S = 1/9^2 < 1/8 - 1/9`
`⇒ S = 1/2^2+1/3^2+1/4^2+...+1/9^2 < 1 - 1/9`
`S = 1/2^2+1/3^2+1/4^2+...+1/9^2 < 8/9 `
` - - - - - - - -- - - -- - -`
`S = 1/2^2+1/3^2+1/4^2+...+1/9^2 < 1/2.3+ 1/3.4 + 1/4.5 + ... + 1/9.10`
`S = 1/2^2+1/3^2+1/4^2+...+1/9^2 < 1/2 - 1/10`
`S = 1/2^2+1/3^2+1/4^2+...+1/9^2 < 4/10 = 2/5`
`⇒ 2/5<S<8/9 (đpcm )`
