Question

chứng minh

1/4+1/16+1/36+...+196 < 1/2

cho đáp án với <333

 

Answer 1

`1/4+1/16+1/36+...+1/196`

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

`= 1/(2^2)*( 1/ + 1/( 2^2 ) + 1/(3^2)+.....+1/(7^2))`

Ta có : `1/(2^2)<1/(1*2)=1-1/2`

`1/(3^2)<1/(2*3)=1/2-1/3`

`.....`

`1/(7^2)<1/(6*7)=1/6-1/7`

Do `1/( 2^2 ) + 1/(3^2)+.....+1/(7^2)<1-1/2+1/2-1/3+.....+1/6-1/7=1-1/7<1`

`=> 1/ + 1/( 2^2 ) + 1/(3^2)+.....+1/(7^2)<2`

`=>  1/(2^2)*( 1/ + 1/( 2^2 ) + 1/(3^2)+.....+1/(7^2))<1/2`

`=>1/4+1/16+1/36+...+1/196<1/2`

Vậy `1/4+1/16+1/36+....+1/196<1/2`