证明1/(n+1)+1/(n+2)+...+1/(n+n)

题目

证明1/(n+1)+1/(n+2)+...+1/(n+n)

答案

可用导数证明:对x > 0,ln(1+x) > x/(1+x).因此ln(k+1)-ln(k) = ln(1+1/k) > (1/k)/(1+1/k) = 1/(k+1).对k = n,n+1,...,2n-1将上式求和即得1/(n+1)+1/(n+2)+...+1/(2n) < ln(2n)-ln(n) = ln(2).