lim(n→无穷)[1/(n^2+1)+2/(n^2+2)+.+2n/(n^2+n)]

来源：学生作业帮助网编辑：作业帮时间：2024/04/30 12:46:16

lim(n→无穷)[1/(n^2+1)+2/(n^2+2)+.+2n/(n^2+n)]
lim(n→无穷)[1/(n^2+1)+2/(n^2+2)+.+2n/(n^2+n)]

lim(n→无穷)[1/(n^2+1)+2/(n^2+2)+.+2n/(n^2+n)]
1/(n^2+1)+2/(n^2+2)+.+n/(n^2+n)
≤1/(n^2+1)+2/(n^2+1)+.+n/(n^2+1)
≤(n^2+n)/2(n^2+1)····（1）
1/(n^2+1)+2/(n^2+2)+.+n/(n^2+n)
≥1/(n^2+n)+2/(n^2+n)+.+n/(n^2+n)
≥(n^2+n)/2(n^2+n)
=1/2
(1)的极限为1/2 根据夹逼定理极限为1/2

...这个题目还有点小问题，最后一项应该是n/(n^2+n) 或者是2n/(n^2+2n)
但是不影响过程
以最后一项是n/(n^2+n)为例
方法是使用夹逼定理
对任意1<=i<=n i/(n^2+n) <=i/(n^2+i)<=i/n^2
lim(1+2+...n)/(n^2+n)<= 原式<=lim (1+2+...n)/n^2
即...

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