求证:n属于正整数,1/(n+1)+1/(n+2)~+1/2n>=2n/3n+1

题目

求证:n属于正整数,1/(n+1)+1/(n+2)~+1/2n>=2n/3n+1

答案

用数学归纳法,当n=1时不等式成立.若结论对n成立,则有1/(n+2)+...+1/2n+1/2n+1+1/(2n+2)>=2n/(3n+1)+1/(2n+1)+1/(2n+2)-1/(n+1)=2n/(3n+1)+1/(2n+1)-1/(2n+2)=2n/(3n+1)+1/(2n+1)(2n+2)>(2n+2)/(3n+4),最后一个不等式是因为(倒推)1/(2n+1)(2n+2)>(2n+2)/(3n+4)-2n/(3n+1),等价于1/(2n+1)(2n+2)>2/(3n+4)(3n+1)等价于9n^2+15n+4>8n^2+12n+4