Ta có:
` S = 2/5 + (2/5)^2 + (2/5)^3 + ... + (2/5)^{2019} + (2/5)^{2020} `
` <=> 5/2 S = 1 + 2/5 + (2/5)^2 + ... + (2/5)^{2018} + (2/5)^{2019} `
` <=> 5/2 S - S = 1 + 2/5 + (2/5)^2 + ... + (2/5)^{2018} + (2/5)^{2019} - 2/5 - (2/5)^2 - (2/5)^3 - ... - (2/5)^{2019} - (2/5)^{2020} `
` <=> 3/2 S = 1 + (2/5 - 2/5) + [(2/5)^2 - (2/5)^2] + ... + [(2/5)^2019 - (2/5)^2019)] - (2/5)^{2020} `
` <=> 3/2 S = 1 - (2/5)^{2020} `
` <=> S = (1 - \frac{2^{2020}}{5^{2020}}) : 3/2 `
` <=> S = 2/3 - \frac{2^{2021}}{3.5^{2020}} `
Do: ` 2/3 - \frac{2^{2021}}{3.5^{2020}} < 2/3 `
` => S < 2/3 ` ` (đpcm) `
