Giải thích các bước giải:
Ta có: `S = 1/4 + 2/4^2 + ... + 2021/4^2021`
`=> 4S = 1 + 2/4 + ... + 2021/4^2020`
`=> 4S - S = (1 + 2/4 + ... + 2021/4^2020) - (1/4 + 2/4^2 + ... + 2021/4^2021)`
`=> 3S = 1 + 1/4 + 1/4^2 + ... + 1/4^2020 - 2021/4^2021`
`=> 3S < 1 + 1/4 + 1/4^2 + ... + 1/4^2020 (1)`
Đặt `A = 1 + 1/4 + 1/4^2 + ... + 1/4^2020`
`=> 4A = 4 + 1 + 1/4 + ... + 1/4^2019`
`=> 4A - A = (4 + 1 + 1/4 + ... + 1/4^2019) - (1 + 1/4 + 1/4^2 + ... + 1/4^2020)`
`=> 3A = 4 - 1/4^2020`
`=> A = 4/3 - 1/4^2020 . 3`
`=> A < 4/3 (2)`
Từ `(1)` và `(2) => 3S < 4/3`
`=> S < 4/9 < 4/8 = 1/2`
`=> S < 1/2 (đpcm)`
