CHỨNG MINH A= 1/4 + 1/16 + 1/36 + 1/64 + 1/100 + 1/144+ 1/196 + 1/256 + 1 /324< 1/2 03/10/2021 Bởi Everleigh CHỨNG MINH A= 1/4 + 1/16 + 1/36 + 1/64 + 1/100 + 1/144+ 1/196 + 1/256 + 1 /324< 1/2
`1/4 + 1/16 + 1/36 + 1/64 + 1/100 + 1/144+ 1/196 + 1/256 + 1 /324` `= 1/4 (1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + 49 + 1/64 + 1/81)` `= 1/4 ( 1 + (1/2)^2 + (1/3)^2 + (1/4)^2 + (1/5)^2 + (1/6)^2 + (1/7)^2 + (1/8)^2 + (1/9)^2 ) ` `= 1/4 . (1 + 1/(1.2) + 1/(2.3) + 1/(3.4) + 1/(4.5) + 1/(5.6) + 1/(6.7) + 1/(6.8) + 1/(7.8) + 1/(8.9) )` `= 1/4 . (1 + 1 – 1/2 + 1/2 – 1/3 + 1/3 – 1/4 + 1/4 – 1/5 +……. – 1/9)` `= 1/4 .( 2 – 1/9)` `= 1/4 . 17/9` `= 17/36 < 18/36 = 1/2` `=> A < 1/2` Bình luận
Đáp án: Giải thích các bước giải: A= 1/4+ 1/16+ 1/36+ 1/64+1/100+ 1/144+ 1/196+ 1/256+ 1/324 = 1/2^2+ 1/4^2+ 1/6^2+ 1/8^2+ 1/10^2+ 1/12^2+ 1/14^2+ 1/16^2+ 1/18^2 = 1/2^2.( 1/1^2+ 1/2^2+ 1/3^2+…..+ 1/9^2) = 1/4. ( 1/1^2+ 1/2^2+ 1/3^2+…..+ 1/9^2) Đặt B= 1/1^2+ 1/2^2+ 1/3^2+…..+ 1/9^2 Vì 1/2^2< 1/1.2 1/3^2< 1/2.3 …… 1/9^2< 1/8.9 => B<1+ 1/1.2+ 1/2.3+..,…+ 1/8.9 =1+ 1- 1/2+ 1/2- 1/3+…..+ 1/8- 1/9 = 2- 1/9< 2( vì 1/9>0) => B<2 => A< 1/4.2= 1/2 => A< 1/2 Vậy A< 1/2 Chúc bạn học tốt! Bình luận
`1/4 + 1/16 + 1/36 + 1/64 + 1/100 + 1/144+ 1/196 + 1/256 + 1 /324`
`= 1/4 (1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + 49 + 1/64 + 1/81)`
`= 1/4 ( 1 + (1/2)^2 + (1/3)^2 + (1/4)^2 + (1/5)^2 + (1/6)^2 + (1/7)^2 + (1/8)^2 + (1/9)^2 ) `
`= 1/4 . (1 + 1/(1.2) + 1/(2.3) + 1/(3.4) + 1/(4.5) + 1/(5.6) + 1/(6.7) + 1/(6.8) + 1/(7.8) + 1/(8.9) )`
`= 1/4 . (1 + 1 – 1/2 + 1/2 – 1/3 + 1/3 – 1/4 + 1/4 – 1/5 +……. – 1/9)`
`= 1/4 .( 2 – 1/9)`
`= 1/4 . 17/9`
`= 17/36 < 18/36 = 1/2`
`=> A < 1/2`
Đáp án:
Giải thích các bước giải:
A= 1/4+ 1/16+ 1/36+ 1/64+1/100+ 1/144+ 1/196+ 1/256+ 1/324
= 1/2^2+ 1/4^2+ 1/6^2+ 1/8^2+ 1/10^2+ 1/12^2+ 1/14^2+ 1/16^2+ 1/18^2
= 1/2^2.( 1/1^2+ 1/2^2+ 1/3^2+…..+ 1/9^2)
= 1/4. ( 1/1^2+ 1/2^2+ 1/3^2+…..+ 1/9^2)
Đặt B= 1/1^2+ 1/2^2+ 1/3^2+…..+ 1/9^2
Vì 1/2^2< 1/1.2
1/3^2< 1/2.3
……
1/9^2< 1/8.9
=> B<1+ 1/1.2+ 1/2.3+..,…+ 1/8.9
=1+ 1- 1/2+ 1/2- 1/3+…..+ 1/8- 1/9
= 2- 1/9< 2( vì 1/9>0)
=> B<2
=> A< 1/4.2= 1/2
=> A< 1/2
Vậy A< 1/2
Chúc bạn học tốt!