S=1/2^2+1/3^2+1/4^2+…+1/9^2 chứng minh rằng 2/5

18/10/2021

By Kaylee

S=1/2^2+1/3^2+1/4^2+…+1/9^2 chứng minh rằng 2/5 { "@context": "https://schema.org", "@type": "QAPage", "mainEntity": { "@type": "Question", "name": " S=1/2^2+1/3^2+1/4^2+...+1/9^2 chứng minh rằng 2/5

0 bình luận về “S=1/2^2+1/3^2+1/4^2+…+1/9^2 chứng minh rằng 2/5<S<8/9”

1. `1/2^2+1/3^2+…+1/9^2<1(1.2)+1/(2.3)+…+1/(8.9)`

   `⇔S<1-1/2+1/2-1/3+…+1/8-1/9`

   `⇔S<1-1/9`

   `⇔S<8/9`

   `1/2^2+1/3^2+…+1/9^2>1/(2.3)+1/(3.4)+…+1/(9.10)`

   `⇔S>1/2-1/3+1/3-1/4+…+1/9-1/10`

   `⇔S>1/2-1/10`

   `⇔S>2/5`

   `⇒2/5<S<8/9`

    

   Trả lời

2. Ta có :

   `1/2^2  < 1 – 1/2`

   `1/3^2 < 1/2  – 1/3`

   `1/4^2 < 1/3 – 1/4`

   `…`

   `S = 1/9^2 < 1/8 – 1/9`

   `⇒ S =  1/2^2+1/3^2+1/4^2+…+1/9^2 < 1 – 1/9`

   `S = 1/2^2+1/3^2+1/4^2+…+1/9^2 < 8/9 `

   ` – – – – – – – —   – – — – -`

   `S = 1/2^2+1/3^2+1/4^2+…+1/9^2 < 1/2.3+  1/3.4 + 1/4.5  + … + 1/9.10`

   `S = 1/2^2+1/3^2+1/4^2+…+1/9^2 < 1/2 – 1/10`

   `S  = 1/2^2+1/3^2+1/4^2+…+1/9^2 <  4/10 = 2/5`

   `⇒ 2/5<S<8/9 (đpcm )`

   Trả lời