Question

已知n是大于1的自然数，求证log n (n+1)>log n+1 (n+2)

急需

Answer 1

logn(n+1)=ln(n+1)/ln(n)={ln(n)+ln[(n+1)/n]}/ln(n)=1+ln[(n+1)/n]/ln(n)
同样logn+1(n+2)=1+ln[(n+2)/(n+1)]/ln(n+1)
(n+1)/n>(n+2)/(n+1) => ln[(n+1)/n]>ln[(n+2)/(n+1)]
又ln(n)<ln(n+1)
所以1+ln[(n+1)/n]/ln(n)>1+ln[(n+2)/(n+1)]/ln(n+1)
则logn(n+1)>logn+1(n+2)