`S = 1/3 - 2/3^2 + 3/3^3 - ... - 100/3^100`
`⇒ 3S = 1 - 2/3 + 3/3^2 - .... - 100/3^99`
`⇒ 3S + S = 1 - 1/3 + 1/3^2 - ... - 1/3^99 - 100/3^100`
Đặt `D = 1 - 1/3 + 1/3^2 - ... - 1/3^99`
`⇒ 3D = 3 - 1 + 1/3 - 1/3^2 + ... - 1/3^98`
`⇒ 4D = 3 - 1/3^99`
`⇒ D = (3 - 1/3^99). 1/4`
Thay `D` vào `S` ta có:
`4C = 1/4. (3 - 1/3^99) - 100/3^100`
`⇒ 4S = 3/4 - 1/(4. 3^99) - 100/3^100`
`⇒ S = 1/4. (3/4 - 1/(4. 3^99) - 100/3^100)`
`⇒ S = 3/16 - 1/(4^2. 3^99) - 25/3^100`
`⇒ S = 3/16 - (1/(4^2. 3^99) + 25/3^100)`
`⇒ S < 3/16` mà `3/16 < 1/5`
`⇒ S < 1/5`