怎么证明logN N+1 乘以logN N-1
题目
怎么证明logN N+1 乘以logN N-1 <1
答案
真数>0,即N+1>0,N-1>0,所以N>1
所以logN X在(0,正无穷)单调递增
当10
所以logN N+1 乘以logN N-1 <1
当N>2时要证logN N+1 乘以logN N-1 <1
即证logN N+1 logN N-1<1/logN N-1=logN-1 N
即证(lgN+1)/lgN即证(lgN+1)(lgN-1)<(lgN)^2
因为ab<=(a+b)^2/4
所以(lgN+1)(lgN-1)<=[(lgN+1)+(lgN-1)]^2/4=lg[(N^2-1)]^2/4
<=[lg(N^2)]^2/4=(lgN)^2,证得
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