`A = 1/3 - 2/3^2 + 3/3^3 - 4/3^4 + ... + 2023/3^2023`

`3A = 1 - 2/3 + 3/3^2 - 4/3^3 + ... + 2023/3^2022`

`3A + A = (1 - 2/3 + 3/3^2 - 4/3^3 + ... + 2023/3^2022) + (1/3 - 2/3^2 + 3/3^3 - 4/3^4 + ... + 2023/3^2023)`

`4A = 1 - 1/3 + 1/3^2 - 1/3^3 + ... + 1/3^2022 + 2023/3^2023`

    Gọi `B = 1 - 1/3 + 1/3^2 - 1/3^3 + ... + 1/3^2022`

           `3B = 3 - 1 + 1/3 - 1/3^2 + ... + 1/3^2021`

           `3B + B = (3 - 1 + 1/3 - 1/3^2 + ... + 1/3^2021) + (1 - 1/3 + 1/3^2 - 1/3^3 + ... + 1/3^2022)`

           `4B = 3 + 1/3^2022`

           `B = 3/4 + 1/(3^2022 . 4)`

  `=> 4A = 3/4 + 1/(3^2022 . 4) - 2023/(3^2023)`

  `=> 4A = 3/4 + (3/(3^2023 . 4) - (2023 . 4)/(3^2023 . 4))`

  `=> 4A = 3/4 + (3 - 8092)/(3^2023 . 4)`

  `=> 4A = 3/4 + (-8089)/(3^2023 . 4)`

  `=> 4A = 3/4 - 8089/(3^2023 . 4)`

  `=> A = 3/16 - 8089/(3^2023 . 16)`

      Vì `3/16 - 8089/(3^2023 . 16) < 3/16` nên:

  `=> A < 3/16`

         Vậy: `A < 3/16`