高二一道数列题数列{An}的通项公式An=1/(n+1)+1/(n+2)+.+1/(n+n),求证{An}为递增数列

问题描述:

高二一道数列题
数列{An}的通项公式An=1/(n+1)+1/(n+2)+.+1/(n+n),求证{An}为递增数列

因为A(n+1)-An=1/(n+1+1)+1/(n+2+1)+.+1/(n+1+n+1)-[1/(n+1)+1/(n+2)+.+1/(n+n)]=1/(2n+1)+1/(2n+2)-1/(n+1)=[(2n+2+2n+1)-2(2n+1)]/[2(n+1)(2n+1)=1/[2(n+1)(2n+1)]>0所以{An}为递增数列