`A = 2 (1/(1.3) + 1/(3.5) + 1/(5.7) + .... + 1/(2017. 2019) )`
` = 2/(1.3) + 2/(3.5) + 2/(5.7) + .... + 2/(2017.2019)`
` = 1 - 1/3 + 1/3 - 1/5 + 1/5 - 1/7 + .... + 1/2017 - 1/2019`
`= 1 - 1/2019`
` = 2018/2019`
``
`A =1/31 +1/32 + 1/33 + .. 1/60`
` = (1/31 +1/32 + 1/33 + .... + 1/39 + 1/40) + (1/41 + 1/42 + 1/43 + .... + 1/49 + 1/50) + (1/51 + 1/52 + 1/53 + .... + 1/60)` (có `3` nhóm, mỗi nhóm gồm `10` số hạng)
Ta thấy :
`1/31 < 1/30`
`1/32 < 1/30`
`1/33 < 1/30`
....
`1/39 <1/30`
`1/40 < 1/30`
`=> 1/31 +1/32 + 1/33 + .... + 1/39 + 1/40 < 1/30 . 10 = 1/3 (1)`
Tương tự ta có :
`1/41 + 1/42 + 1/43 + .... + 1/49 + 1/50 <1/4 (2)`
`1/51 + 1/52 + 1/53 + .... + 1/60 < 1/5 (3)`
Từ `(1) ; (2) ; (3)` suy ra `S < 1/3 + 1/4 + 1/5`
`=> S < 47/60`
`=> S < 48/60 `
`=> S < 4/5`
