Chứng minh rằng : 1/2^2 + 1/4^2 + 1/6^2 +....+1/100^2 < 1/2 Giúp mình ạ! Cảm ơn các bạn!

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2 năm trước

Loga Sinh Học lớp 12

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nguyenthihuong123tp

Đáp án:

Giải thích các bước giải:

`đặt S=1/2^2 + 1/4^2 + 1/6^2 +....+1/100^2`

`2S<2/1.3+2/3.5+2/5.7......+2/99.101`

`2S<1-1/3+1/3-1/5+1/5-1/7+.....+1/99-1/101`

`2S<1-1/101`

`2S<100/101`

`S<50/101<1/2`

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ntramy983

`A= 1/(2^2) + 1/(4^2)+ 1/(6^2)+...+ 1/(100^2)`

`A=1/(2^2 .1) + 1/(2^2 . 2^2) + 1/(2^2 . 2^3) +...+ 1/(2^2 . 50^2)`

`A= 1/(2^2) . ( 1/1 + 1/(2^2) + 1/(3^2) +...+ 1/(50^2))`

`=> 1/(2^2) + 1/(3^2) +...+ 1/(50^2) < 1/(1.2)+ 1/(2.3)+...+ 1/(49.50)``

`=> A < 1/(2^2)(1/(1.2) + 1/(2.3) +...+ 1/(49.50))`

`=> A < 1/4 ( 1+ 1 - 1/2 + 1/2 - 1/3 +...+ 1/49 - 1/50)`

`=> A < 1/4 ( 1+ 1 - 1/50)`

`=> A < 1/4 ( 2 - 1/50)` mà `2 - 1/50 < 2`

`=> A < 1/4 .2`

`=> A < 1/2 ( đpcm)`

Vote (0) Phản hồi (0) 2 năm trước

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