一道数列证明题设各项为正的数列{an}满足A1=1,A(n+1)=lnAn+An+2,证明An≤2∧n -1说明A(n+1)为下角标的n+1即第n+1项,2∧n 2的n次
问题描述:
一道数列证明题
设各项为正的数列{an}满足A1=1,A(n+1)=lnAn+An+2,证明An≤2∧n -1
说明A(n+1)为下角标的n+1即第n+1项,2∧n 2的n次
答
证明:首先考察f(x)=lnx-(x-1)(x>=1)
f(x)'=1/x-1=(1-x)/x当x>=1时f(x)lnx-(x-1)lnx1
=>A(n+1)=ln(A(n))+A(n)+2A(n+1)+1