Mong mn giúp mình vs ạ, Thanks mn nhiều

Đáp án:

`B13: A = 1/(1.21) + 1/(2 . 22) + 1/(3.23) + .... + 1/(80.100) ; B = 1/(1.81) + 1/(2.82) + ..... + 1/(20.100).` Tính `A/B`

`A = 1/(1.21) + 1/(2 . 22) + 1/(3.23) + .... + 1/(80.100)`

   `= 1/20 (1 - 1/21 + 1/2 - 1/22 + 1/3 - 1/33 + ...... + 1/80 - 1/100)`

   `= 1/20 [(1 + 1/2 + 1/3 + ... + 1/80) - (1/21 + 1/22 + .... + 1/100)]`

`B = 1/(1.81) + 1/(2.82) + ..... + 1/(20.100)`

   `= 1/80 [(1 + 1/2 + 1/3 + .... + 1/20) - (1/81 + 1/82 + .... + 1/100)]`

`A/B = (1/20)/(1/80) = 1/20 . 80 = 4`

`B14: S = 1/3 - 2/(3^2) + 3/(3^3) - 4/(3^4) + ...... + 99/(3^99) - 100/(3^100) .` So sánh `S` và `1/5`

`3S = 1 - 2/3 + 3/(3^2) - 4/(3^3) + ..... - 100/(3^99)`

`3S + S = (1 - 2/3 + 3/(3^2) - 4/(3^3) + ..... - 100/(3^99)) + (1/3 - 2/(3^2) + 3/(3^3) - 4/(3^4) + ...... + 99/(3^99) - 100/(3^100))`

`4S = 1 - 1/100`

`S = (3^100 - 1)/(4.3^100)`

`= (4.3^100 - 1)/(4.3^100)`

`= 1 - 1/(4.3^100)`

  `1/5 > 1/(4.3^100) ⇒ 1 - 1/(4.3^100) < 1/5 (đpcm)`