Đặt `A = 1/2 + 2/2^2 + 3/2^3 + 4/2^4 + ... + 2021/2^2021`
`⇒ 2A = 2. (1/2 + 2/2^2 + 3/2^3 + 4/2^4 + ... + 2021/2^2021)`
`⇒ 2A = 1 + 1 + 3/2^2 + 4/2^3 + ... + 2021/2^2020`
`⇒ 2A - A = (1 + 1 + 3/2^2 + 4/2^3 + ... + 2021/2^2020) - (1/2 + 2/2^2 + 3/2^3 + 4/2^4 + ... + 2021/2^2021)`
`⇒ A = 1 + (1 - 1/2) + (3/2^2 - 2/2^2) + (4/2^3 - 3/2^3) + ... + (2021/2^2020 - 2020/2^2020) - 2021/2^2021`
`⇒ A = 1 + 1/2 + 1/2^2 + 1/2^3 + ... + 1/2^2020 - 2021/2^2021`
`⇒ 2A = 2. (1 + 1/2 + 1/2^2 + 1/2^3 + ... + 1/2^2020 - 2021/2^2021)`
`⇒ 2A = 2 + 1 + 1/2 + 1/2^2 + ... + 1/2^2019 - 2021/2^2020`
`⇒ 2A - A = (2 + 1 + 1/2 + 1/2^2 + ... + 1/2^2019 - 2021/2^2020) - (1 + 1/2 + 1/2^2 + 1/2^3 + ... + 1/2^2020 - 2021/2^2021)`
`⇒ A = 2 + (1 - 1) + (1/2 - 1/2) + (1/2^2 - 1/2^2) + ... + (1/2^2019 - 1/2^2019) - (2021/2^2020 - 1/2^2020) - 2021/2^2021`
`⇒ A = 2 - 2020/2^2020 - 2021/2^2021 < 2`
`⇒ 1/2 + 2/2^2 + 3/2^3 + 4/2^4 + ... + 2021/2^2021 < 2`
`⇒ đpcm`
