Đáp án:
$\\$
Đặt `A=1/2^2 + 1/3^2 + 1/4^2 + ... + 1/2021^2`
`-> A = 1/4 + 1/3^2 + 1/4^2 + ... + 1/2021^2`
$\bullet$ `1/3^2 + 1/4^2 + ... + 1/2021^2`
Vì `3.3 > 2.3`
`-> 1/3^2 < 1/(2.3)`
Tương tự có :
`1/4^2 < 1/(3.4)`
`...........`
`1/2021^2 < 1/(2020.2021)`
Cộng theo vế ta được :
`-> 1/3^2 +1/4^2 + ... + 1/2021^2 < 1/(2.3) + 1/(3.4) + ... + 1/(2020 . 2021)`
`-> 1/3^2 + 1/4^2 + ... + 1/2021^2 < 1/2 - 1/3 +1/3 - 1/4 + ... + 1/2020 - 1/2021`
`-> 1/3^2 + 1/4^2 + ... + 1/2021^2 < 1/2 + (-1/3 + 1/4) + ... + (-1/2020 + 1/2020) - 1/2021`
`-> 1/3^2 + 1/4^2 + ... + 1/2021^2 < 1/2 - 1/2021`
`-> 1/4 + 1/3^2 + 1/4^2 + ... + 1/2021^2 < 1/4 + 1/2 - 1/2021`
`-> A < 3/4 - 1/2021`
Ta thấy : `3/4 - 1/2021 < 3/4`
`-> A < 3/4 - 1/2021 < 3/4`
`-> A<3/4` (đpcm)
