求证:ln(n+1)>1/3+1/5+1/7+·······+1/(2n+1) (n∈N)

问题描述:

求证:ln(n+1)>1/3+1/5+1/7+·······+1/(2n+1) (n∈N)

因为1/(2x+1)是凹函数,所以1/3+1/5+...1/(2n+1)因为lnx是凸函数,所以[ln1+ln(2n+1)]/21/3然后假设ln(n+1)>1/3+1/5+...+1/(2n+1)只要证ln(n+2)>1/3+1/5+...+1/(2n+1)+1/(2n+3)因为1/3+1/5+...+1/(2n+1)+1/(2n+3)ln(n+1)+1/(2n+3)只要证ln(1+1/(n+1))>1/(2n+3)只要证1+1/(n+1)>e^[1/(2n+3)]两边同时n+1次方,有e>e^0.5,成立