Đặt :` B = 1/3 + 2/3^2 + 3/3^3 + .... + 2019/3^2019`
`⇒ 3B = 1 + 2/3 + .... + 2019/3^2018`
`⇒ 3B - B = ( 1 + 2/3 + ... + 2019/3^2018 ) - ( 1/3 + 2/3^2 + .... + 2019/3^2019 )`
`⇒ 2B = 1 + 1/3 + .... + 1/3^2018 - 2019/3^2019`
`⇒ 6B = 3 + 1 + 1/3 + .... + 1/3^2019 - 2019/3^2018`
`⇒ 6B - 2B = ( 3 + 1 + 1/3 + ... + 1/3^2019 - 2019/3^2018 ) - ( 1 + 1/3 + ... + 1/3^2018 - 2019/3^2019 )`
`⇒ 4B = 3 - 2020/3^2018 + 2019/3^2019`
`⇒ B = 3/4 - 4041/3^2019 < 3/4`
`⇒ B < 3/4` ( Điều phải chứng minh )