Question

lim(n→∞){n/（n^2+1）+n/（n^2+2^2）+....+n/（n^2+n^2）}求此式的极限？说明一下那是n除以n的平方加1,

Answer 1

简单计算一下即可，答案如图所示

Answer 2

lim(n→∞)n[1/(n^2+1)+1/(n^2+2)+1/(n^2+3)+……+1/(n^2+n)]
lim(n→∞)[n/(n^2+1)+n/(n^2+2)+n/(n^2+3)+……+n/(n^2+n)]
对于每一项来说，分子均为一次项，分母为二次项，均趋向0
lim(n→∞)n[1/(n^2+1)+1/(n^2+2)+1/(n^2+3)+……+1/(n^2+n)]=0

Answer 3

原式=lim(n→∞)[(1/n)/(1+(1/n)^2)+(1/n)/(1+(2/n)^2)+...+(1/n)/(1+(n/n)^2)]
（每项上下同时除以n^2）
=lim(n→∞)1/n*[1/(1+(1/n)^2)+1/(1+(2/n)^2)+...+1/(1+(n/n)^2)]
=∫(0→1)1/(1+x^2)dx
（区间[0,1]的分点为i/n）
=arctanx|(0→1)
=π/4