Cho S= 1/3 - 2/3^2 + 3/3^3 - 4/3^4 +...+ 99/3^99 - 100/3^100. So sánh S và 1/5

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

`3S = 1 - 2/3 + 3/3^2 - 4/3^3 + ... + 99/3^98 - 100/3^99`

`3S + S = ( 1 - 2/3 + 3/3^2 - 4/3^3 + ... + 99/3^98 - 100/3^99 ) + ( 1/3 - 2/3^2 + 3/3^3 - 4/3^4 + ... + 99/3^99 - 100/3^100 )`

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

`12S = 3 - 1 + 1/3 - 1/3^2 + 1/3^3 - ... - 1/3^98 + 1/3^99`

`12S + 4S = ( 3 - 1 + 1/3 - 1/3^2 + 1/3^3 - ... - 1/3^98 + 1/3^99 ) + ( 1 - 1/3 +1/3^2 - 1/3^3 + 1/3^4 - ... - 1/3^99 + 1/3^100 )`

`16S = 3 - 99/3^99 - 100/3^100`

`S = (3-99/3^99 - 100/3^100)/16`

mà `(3- 99/3^99 - 100/3^100)/16 < 3/15 = 1/5`

`=> S < 1/5`

       vậy `S<1/5`