`A = 1/2^2 + 1/3^2 + 1/4^2 + ... + 1/2020^2`
Ta có:
`1/2^2 = 1/(2. 2) < 1/(1. 2)`
`1/3^2 = 1/(3. 3) < 1/(2. 3)`
`1/4^2 = 1/(4. 4) < 1/(3. 4)`
`...`
`1/2020^2 = 1/(2020. 2020) < 1/(2019. 2020)`
`=> 1/2^2 + 1/3^2 + 1/4^2 + ... + 1/2020^2 < 1/(1. 2) + 1/(2. 3) + 1/(3. 4) + ... + 1/(2019. 2020)`
`=> A < 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/2019 - 1/2020`
`=> A < 1 - 1/2020`
Vì `1/2020 > 0`
`<=> - 1/2020 < 0`
`<=> 1 - 1/2020 < 1`
`<=> A < 1 - 1/2020 < 1`
`<=> A < 1(đpcm)`
