设n为大于1的自然数,求证:[1/(n+1)]+[1/(n+2)]+[1/(n+3)]+...+1/...
发布网友
共2个回答
热心网友
1/(n+1)>1/2n, 后面的各项都大于1/2n, 总共有n项, 所以
1/(n+1)+[1/(n+2)]+[1/(n+3)]+...+1/(2n)>n*[1/(2n)]=n/(2n)=1/2
热心网友
[1/(n+1)] + [1/(n+2)]+[1/(n+3)]+...+1/(2n)
> [1/(n+1)] + [1/(2n)]+[1/(2n)]+...+1/(2n)
= [1/(n+1)] + [(n-1)/(2n)]
= (2n+n^2-1) / [(2n)(n+1)]
= (n+n^2) / [(2n)(n+1)] + (n-1) / [(2n)(n+1)]
= 1/2 + (n-1) / [(2n)(n+1)]
> 1/2
