运用柯西不等式证明:4/7
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运用柯西不等式证明:
4/7
4/7
![运用柯西不等式证明:4/7](/uploads/image/z/16119521-17-1.jpg?t=%E8%BF%90%E7%94%A8%E6%9F%AF%E8%A5%BF%E4%B8%8D%E7%AD%89%E5%BC%8F%E8%AF%81%E6%98%8E%EF%BC%9A4%2F7)
证明:先证明左边,利用柯西不等式
(1/(n+1)+1/(n+2)+...+1/2n)(n+1+n+2+...2n)>=(1+1...+1)^2=n^2
=>(1/(n+1)+1/(n+2)+...+1/2n)>=n^2/((3n+1)2n/2)=2n/(3n+1)=2/(3/2+1/n)
显然在n=2时2/(3/2+1/n)取最小值,故2n/(3n+1)>=4/7
当且仅当1/(n+1)=1/(n+2)...1/2n且n=2取等号,显然是取不到的,故有
4/7
(1/(n+1)+1/(n+2)+...+1/2n)(n+1+n+2+...2n)>=(1+1...+1)^2=n^2
=>(1/(n+1)+1/(n+2)+...+1/2n)>=n^2/((3n+1)2n/2)=2n/(3n+1)=2/(3/2+1/n)
显然在n=2时2/(3/2+1/n)取最小值,故2n/(3n+1)>=4/7
当且仅当1/(n+1)=1/(n+2)...1/2n且n=2取等号,显然是取不到的,故有
4/7