`M = 3/2^3 + 4/3^3 + 5/4^3 + ... + 2022/2021^3`

Nhận xét:

`3/2^3>2/2^3=1/2^2>1/2.3`

`4/3^3>3/3^3=1/3^2>1/3.4`

`5/4^3>4/4^3=1/4^2>1/4.5`

`...`

`2022/2021^3>2021/2021^3=1/2021^2>1/2021.2022`

`=>M>1/2.3+1/3.4+1/4.5+...+1/2021.2022`

`=>M>1/2-1/3+1/3-1/4+1/4-1/5+...+1/2021-1/2022`

`=>M>1/2-1/2022`

`=>M>505/1011`

Vậy `M>505/1011`(đpcm)