cho A= 1/2^2+1/3^2+1/4^2+…+1/9^2. Chứng minh 2/5 01/07/2021 Bởi Adeline cho A= 1/2^2+1/3^2+1/4^2+…+1/9^2. Chứng minh 2/5 { "@context": "https://schema.org", "@type": "QAPage", "mainEntity": { "@type": "Question", "name": " cho A= 1/2^2+1/3^2+1/4^2+...+1/9^2. Chứng minh 2/5
Ta có: `1/2^2 < 1/1.2 ; 1/3^2 < 1/2.3; ….. ; 1/9^2 < 1/8.9` `=> 1/2^2 + 1/3^2 + 1/4^2+………+ 1/9^2 < 1/1.2 + 1/2.3 + …. + 1/8.9` `=> A < 1/1 -1/2 + 1/2 -1/3 +….+ 1/8- 1/9` `=>A < 1/1 -1/9` `=> A < 8/9 (1)` Lại có: `1/2^2 > 1/2.3 ; 1/3^2 > 1/3.4 ; ………; 1/9^2 > 1/9.10` `=> 1/2^2 + 1/3^2 + 1/4^2+………+ 1/9^2 > 1/2.3 + 1/3.4 +…+ 1/9.10` `=> A > 1/2 -1/3 + 1/3 -1/4 +….+ 1/9-1/10` `=> A > 1/2 -1/10 = 2/5 (2)` Từ (1) và (2) `=> 8/9 > A > 2/5` Vậy `8/9 > A > 2/5` Bình luận
Xin hay nhất
Ta có: `1/2^2 < 1/1.2 ; 1/3^2 < 1/2.3; ….. ; 1/9^2 < 1/8.9`
`=> 1/2^2 + 1/3^2 + 1/4^2+………+ 1/9^2 < 1/1.2 + 1/2.3 + …. + 1/8.9`
`=> A < 1/1 -1/2 + 1/2 -1/3 +….+ 1/8- 1/9`
`=>A < 1/1 -1/9`
`=> A < 8/9 (1)`
Lại có: `1/2^2 > 1/2.3 ; 1/3^2 > 1/3.4 ; ………; 1/9^2 > 1/9.10`
`=> 1/2^2 + 1/3^2 + 1/4^2+………+ 1/9^2 > 1/2.3 + 1/3.4 +…+ 1/9.10`
`=> A > 1/2 -1/3 + 1/3 -1/4 +….+ 1/9-1/10`
`=> A > 1/2 -1/10 = 2/5 (2)`
Từ (1) và (2) `=> 8/9 > A > 2/5`
Vậy `8/9 > A > 2/5`