神马作文网放缩法的不等式证明技巧
来源：学生作业帮 编辑：神马作文网作业帮 分类：数学作业 时间：2024/11/11 19:36:46
放缩法的不等式证明技巧
放缩法的不等式证明技巧,我举几例子给你看看：
1、3/2*5/4*7/6*9/8*...2n+1/2n>√（n+1）
3/2*5/4*7/6*9/8*...2n+1/2n）²
=（3/2*5/4*7/6*9/8*...2n+1/2n）（3/2*5/4*7/6*9/8*...2n+1/2n）
>(3/2*5/4*7/6*9/8*...2n+1/2n)(4/3*6/5*8/7*10/9...(2n+2)(2n+1))
=(2n+2)/2=n+1
∴ 3/2*5/4*7/6*9/8*...2n+1/2n>√(n+1)
2、证明|sin(x-y)|≤|sin(x-z)|+|sin(z-y)|
∵|cosx|≤1
|sin(x-y)|=|sin(x-y-z+z)|=|sin[(x-z)-(y-z)]|
=|sin[(x-z)cos(y-z)-cos(x-z)sin(y-z)|
=|sin[(x-z)cos(y-z)+cos(x-z)sin(z-y)|
≤|sin[(x-z)cos(y-z)|+|cos(x-z)sin(z-y)|
≤|sin[(x-z)|+|sin(z-y)|
1、3/2*5/4*7/6*9/8*...2n+1/2n>√（n+1）
3/2*5/4*7/6*9/8*...2n+1/2n）²
=（3/2*5/4*7/6*9/8*...2n+1/2n）（3/2*5/4*7/6*9/8*...2n+1/2n）
>(3/2*5/4*7/6*9/8*...2n+1/2n)(4/3*6/5*8/7*10/9...(2n+2)(2n+1))
=(2n+2)/2=n+1
∴ 3/2*5/4*7/6*9/8*...2n+1/2n>√(n+1)
2、证明|sin(x-y)|≤|sin(x-z)|+|sin(z-y)|
∵|cosx|≤1
|sin(x-y)|=|sin(x-y-z+z)|=|sin[(x-z)-(y-z)]|
=|sin[(x-z)cos(y-z)-cos(x-z)sin(y-z)|
=|sin[(x-z)cos(y-z)+cos(x-z)sin(z-y)|
≤|sin[(x-z)cos(y-z)|+|cos(x-z)sin(z-y)|
≤|sin[(x-z)|+|sin(z-y)|