## cho S = 1/5 mũ 2+ 1/6 mũ 2+……..+1/2020 mũ 2

Question

cho S = 1/5 mũ 2+ 1/6 mũ 2+……..+1/2020 mũ 2

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3 tuần 2021-11-21T17:15:16+00:00 2 Answers 3 views 0

1. 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)

2. Cách làm

ta thấy

1/5^2<1/4.5

1/6^2<1/5.6

……………………

1/2020^2<1/2019.2020

=> 1/5^2+1/6^2+…+1/2020^2<1/4.5+1/5.6+1/2019.2020

=1/4-1/5+1/5-1/6+…+1/2019-1/2020

=1/4-1/2020

lại có

1/5^2>1/5.6

1/6^2>1/6.7

…………………….

1/2020^2>1/2020.2021

mk dang bận nên chỉ làm đc đến đây thôi chúc bạn học tốt