so sánh.1/2+1/2²+1/2³+…+1/2²⁰ và B -212121/-202020 10/08/2021 Bởi Gabriella so sánh.1/2+1/2²+1/2³+…+1/2²⁰ và B -212121/-202020
Tham khảo Đặt `A=\frac{1}{2}+\frac{1}{2^2}+…+\frac{1}{2^{20}}` `⇒2A=1+\frac{1}{2}+…+\frac{1}{2^{19}}` `⇒2A-A=1+\frac{1}{2}+…+\frac{1}{2^{19}}-(\frac{1}{2}+\frac{1}{2^2}+…+\frac{1}{2^{20}})` `⇒A=1-\frac{1}{2^{20}}<1`(*) Mà `B=\frac{-212121}{-202020}=\frac{-21}{-20}=\frac{21}{20}>1`(**) Từ (*),(**)`⇒A<B` `\text{©CBT}` Bình luận
Đặt `A = 1/2 + 1/2^2 + 1/2^3 + … + 1/2^20` `2 . A = 1 + 1/2 + 1/2^2 + … + 1/2^19` `2 . A – A = ( 1 + 1/2 + 1/2^2 + … + 1/2^19 ) – ( 1/2 + 1/2^2 + 1/2^3 + … + 1/2^20 )` `A = 1 – 1/2^20` `A = ( 2^20 – 1 )/2^20` Ta có : B = `( – 212121 )/( – 202020 )` `B = 212121/202020` `B = 21/20` Vì `( 2^20 – 1 )/2^20 < 1 < 21/20 ⇒ A < B` Vậy , `A < B ` Bình luận
Tham khảo
Đặt `A=\frac{1}{2}+\frac{1}{2^2}+…+\frac{1}{2^{20}}`
`⇒2A=1+\frac{1}{2}+…+\frac{1}{2^{19}}`
`⇒2A-A=1+\frac{1}{2}+…+\frac{1}{2^{19}}-(\frac{1}{2}+\frac{1}{2^2}+…+\frac{1}{2^{20}})`
`⇒A=1-\frac{1}{2^{20}}<1`(*)
Mà `B=\frac{-212121}{-202020}=\frac{-21}{-20}=\frac{21}{20}>1`(**)
Từ (*),(**)`⇒A<B`
`\text{©CBT}`
Đặt `A = 1/2 + 1/2^2 + 1/2^3 + … + 1/2^20`
`2 . A = 1 + 1/2 + 1/2^2 + … + 1/2^19`
`2 . A – A = ( 1 + 1/2 + 1/2^2 + … + 1/2^19 ) – ( 1/2 + 1/2^2 + 1/2^3 + … + 1/2^20 )`
`A = 1 – 1/2^20`
`A = ( 2^20 – 1 )/2^20`
Ta có : B = `( – 212121 )/( – 202020 )`
`B = 212121/202020`
`B = 21/20`
Vì `( 2^20 – 1 )/2^20 < 1 < 21/20 ⇒ A < B`
Vậy , `A < B `