作业帮求极限 lim n[1/(n^2+1)+1/(n^2+2^2）+……+1/(n^n+n^n)] (n趋向于无穷大,n^n
来源：学生作业帮 编辑：作业帮 分类：数学作业 时间：2024/04/30 00:32:29
求极限 lim n[1/(n^2+1)+1/(n^2+2^2）+……+1/(n^n+n^n)] (n趋向于无穷大,n^n表示n
=lim n^2·[1/(n^2+1)+1/(n^2+2^2）+……+1/(n^n+n^n)] /n
=lim [n^2/(n^2+1)+n^2/(n^2+2^2）+……+n^2/(n^n+n^n)]·(1/n)
=lim [1/(1+(1/n)^2) +1/(1+(2/n)^2) +……+ 1/(1+(n/n)^2))]·(1/n)
=∫《x从0到1》1/(1+x²) dx
=arctanx |《x从0到1》
=arctan1 - arctan0
=π/4
=lim [n^2/(n^2+1)+n^2/(n^2+2^2）+……+n^2/(n^n+n^n)]·(1/n)
=lim [1/(1+(1/n)^2) +1/(1+(2/n)^2) +……+ 1/(1+(n/n)^2))]·(1/n)
=∫《x从0到1》1/(1+x²) dx
=arctanx |《x从0到1》
=arctan1 - arctan0
=π/4
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