3Cn=(n+2)/n+n/(n+2),证:2n
问题描述:
3Cn=(n+2)/n+n/(n+2),证:2n
答
先证左边:Cn=(n+2)/n+n/(n+2)=1+2/n+n/(n+2)=1+(n^2+2n+4)/(n^2+2n)∴不论n取何值,Cn恒>2∴C1+C2+C3+•••+Cn>2+2+2+•••+2=2n,左边得证
再证右边:由证左边时对Cn的化简得:Cn=2+4/(n^2+2n)
裂项得:Cn=2+2*[1/n-1/(n+2)]
∴C1+C2+C3+•••+Cn=2n+2*[1-1/3+1/2-1/4+1/3-1/5+…+1/(n-1)-1/(n+1)+1/n-1/(n+2)]=2n+2*[1+1/2-1/(n+1)-1/(n+2)]恒<2n+2*(1+1/2)=2n+3,右边得证
综上所述:2n