Ta có S = 1/2² + 1/3² + 1/4² +.....+ 1/9²
< 1/2² + 1/2 × 3 + 1/3 × 4 +.....+ 1/8 × 9
= 1/2² + 1/2 - 1/3 + 1/3 - 1/4+.....+1/8 - 1/9
= 1/4 + 1/2 - 1/9
= 23/36 < 32/36 = 8/9 (1)
Ta lại có S = 1/2² + 1/3² + 1/4² +.....+ 1/9²
> 1/2² + 1/3 × 4 + 1/4 × 5+......+ 1/9 × 10
= 1/2² + 1/3 - 1/4 + 1/4 - 1/5 +......+ 1/9 - 1/10
= 1/2² + 1/3 - 1/10
= 19/20 > 8/20 = 2/5 (2)
Từ (1)(2) cho ta đpcm.
