`A = 1/2^2 + 1/4^2 + 1/6^2 + ... + 1/100^2`

`A = 1/(2 . 2) + 1/(4 . 4) + 1/(6 . 6) + ... + 1/(100 . 100)`

Nhận xét:

      `1/(2. 2) < 1/(1 . 3)`

      `1/(4 . 4) < 1/(3 . 5)`

      `1/(6 . 6) < 1/(5 . 7)`

      `...`

      `1/(100 . 100) < 1/(99 . 101)`

`=> A < 1/(1 . 3) + 1/(3 . 5) + 1/(5 . 7) + ... + 1/(99 . 101)`

`=> A < 1/2 . (2/(1 . 3) + 2/(3 . 5) + 2/(5 . 7) + ... + 2/(99 . 101))`

`=> A < 1/2 . (1 - 1/3 + 1/3 - 1/5 + 1/5 - 1/7 + ... + 1/99 - 1/101)`

`=> A < 1/2 . (1 - 1/101)`

`=> A < 1/2 . 100/101`

     Mà `1/2 < 1 ; 100/101 < 1` nên `1/2 . 100/101 < 1/2`

`=> A < 1/2 . 1/100/101 < 1/2`

`=> A < 1/2`

           Vậy: `A < 1/2`

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