Ta có:

`A = 1/5 + 2/(5^2) + 3/(5^3) + 4/(5^4) + ... + 2023/(5^2023)`

`1/5 A = 1/(5^2) + 2/(5^3) + 3/(5^4) + 4/(5^5) + ... + 2023/(5^2024)`

`A - 1/5 A = 1/5 + 1/(5^2) + 1/(5^3) + 1/(5^4) + ... + 1/(5^2023) - 2023/(5^2024)`

`4/5 A = 1/5 + 1/(5^2) + 1/(5^3) + 1/(5^4) + ... + 1/(5^2023) - 2023/(5^2024)`

`A = (1/5 + 1/(5^2) + 1/(5^3) + 1/(5^4) + ... + 1/(5^2023) - 2023/(5^2024)) : 4/5`

Đặt `B = 1/5 + 1/(5^2) + 1/(5^3) + 1/(5^4) + ... + 1/(5^2023)`

`5B = 1 + 1/5 + 1/(5^2) + 1/(5^3) + ... + 1/(5^2022)`

`5B - B = 1 - 1/(5^2023)`

`4B = 1 - 1/(5^2023)`

`B = (1 - 1/(5^2023)) : 4`

`B = 1/4 - 1/(5^2023 * 4)`

Từ đó ta có:

`A = (1/4 - 1/(5^2023 * 4) - 2023/(5^2024)) : 4/5`

`= 5/16 - 1/(5^2022 * 16) - 2023/(5^2023 * 4) < 5/16`

Vậy `A < 5/16` (đpcm)

`\ttcolor{cyan}{@F4bat on}`