高一 数学 奇妙的不等式 请详细解答,谢谢! (27 16:50:6)

简单 放缩 裂项
1+1/4+1/9+.....+1/81
=1+1/(2*2)+1/(3*3)+....+1/(9*9)
<1+1/(1*2)+1/(2*3)+....+1/(8*9)
=1+1-1/2+1/2-1/3+....+1/8-1/9 消项
=2-1/9
=17/9

先证左边的不等式：
1+1/4+1/9+......+1/81
=1+1/(2*2)+1/(3*3)+...+1/(9*9)
>1+1/(2*3)+1/(3*4)+...+1/(9*10)
=1+(1/2-1/3)+(1/3-1/4)+...+(1/9-1/10)
=1+1/2-1/10
=7/5

再证右边的不等式：
1+1/4+1/9+......+1/81
=1+1/(2*2)+1/(3*3)+...+1/(9*9)
<1+1/(1*2)+1/(2*3)+...+1/(8*9)
=1+(1-1/2)+(1/2-1/3)+...+(1/8-1/9)
=1+1-1/9
=17/9

证毕

简单 放缩 裂项
1+1/4+1/9+.....+1/81
=1+1/(2*2)+1/(3*3)+....+1/(9*9)
<1+1/(1*2)+1/(2*3)+....+1/(8*9)
=1+1-1/2+1/2-1/3+....+1/8-1/9 消项
=2-1/9
=17/9
左边就是分母放大一点。同学，要是学东西左边自己搞掂

中间那个式子< 1+1/1*2+1/2*3+1/3*4 +...+1/8*9=1+1-1/2+1/2-1/3+1/3-1/4+....+1/8-1/9= 1+1-1/9=17/9
中间那个式子>1+1/2*3+1/3*4+...+1/9*10=1+1/2-1/3+1/3-1/4+..+1/9-1/10
=1+1/2-1/10=7/5

原式=1+1/2&