Ta có:
` 1/5^2 < \frac{1}{4.5} `
` 1/6^2 < \frac{1}{5.6} `
` ....... `
` 1/2020^2 < \frac{1}{2019.2020} `
` => 1/5^2 + 1/6^2 + ... + 1/2020^2 < \frac{1}{4.5} + \frac{1}{5.6} + ... + \frac{1}{2019.2020} `
` => S < \frac{1}{4.5} + \frac{1}{5.6} + ... + \frac{1}{2019.2020} `
` => S < 1/4 - 1/5 + 1/5 - 1/6 + ... + 1/2019 . 1/2020 `
` => S < 1/4 - 1/2020 `
Do: ` 1/4 - 1/2020 < 1/4 `
` => S < 1/4 ` (1)
Ta có:
` 1/5^2 > \frac{1}{5.6} `
` 1/6^2 > \frac{1}{6.7} `
` ........... `
` 1/2020^2 > \frac{1}{2020.2021} `
` => 1/5^2 + 1/6^2 + ... + 1/2020^2 > \frac{1}{5.6} + \frac{1}{6.7} + ... + \frac{1}{2020.2021} `
` => S > \frac{1}{5.6} + \frac{1}{6.7} + ... + \frac{1}{2020.2021} `
` => S > 1/5 - 1/6 + 1/6 - 1/7 + ... + 1/2020 - 1/2021 `
` => S > 1/5 - 1/2021 `
Do: ` 1/5 - 2021 > 1/6 `
` => S > 1/6 ` (2)
Từ (1) và (2):
` => 1/6 < S < 1/4 ` ` (đpcm) `
