Đặt `A= 1/2^3 + 1/3^3 + 1/4^3+...+1/2017^3 `
Ta có: `1/2^3 = 1/(2.2.2) < 1/(1.2.3)`
`1/3^3 = 1/(3.3.3) < 1/(2.3.4)`
`......................................................................`
`1/2017^3 = 1/(2017.2017.2017) < 1/(2016.2017.2018)`
`=> 1/2^3 + 1/3^3 + 1/4^3+...+1/2017^3 < 1/(1.2.3) + 1/(2.3.4) +...+1/(2016.2017.2018)`
`=> A < 1/2(1/1.2 - 1/2.3 + 1/2.3 - 1/3.4 +...+1/2016.2017 - 1/2017.2018`
`=> A< 1/2( 1/1.2 - 1/2017.2018)`
`=> A < 1/2 ( 1/2 - 1/2017.2018)`
`=> A < 1/4 - 1/(2.2017.2018) < 1/4 = 1/2^2`
Vậy `1/2^3 + 1/3^3 + 1/4^3+...+1/2017^3 < 1/2^2`
