题目

求高手求极限 要有过程lim(n→∞)〖1/(n^2+1)+2/(n^2+2)+3/(n^2+3)+~+k/(n^2+k)+~〗
可能要用到定积分的定义,5

提问时间：2021-03-28

答案

设[1/(n^2+1)+2/(n^2+2)+.+n/(n^2+n)]=S
则S<[1/(n^2+1)+2/(n^2+1)+.+n/(n^2+1)]=(1+2+……+n）/(n^2+1)=n*(n+1)/2(n^2+1)
因为S>[1/(n^2+n)+2/(n^2+n)+.+n/(n^2+1)]=n*(n+1)/2(n^2+n)
且limn*(n+1)/2(n^2+1)=1/2(n趋于无穷)
limn*(n+1)/2(n^2+n)=1/2(n趋于无穷)
lim[1/(n^2+1)+2/(n^2+2)+.+n/(n^2+n)] n趋于无穷=1/2