lim（n趋向无穷）（n/(n^2+1^2)+n/(n^2+2^2+.n/(n^2+n^2)表示为定积分最好有详尽过程，不要只是个答案

Question

lim（n趋向无穷）（n/(n^2+1^2)+n/(n^2+2^2+.n/(n^2+n^2)
表示为定积分
最好有详尽过程，不要只是个答案

Answer 1

原式=lim(n->∞){(1/n)[n²/(n²+1²)+n²/(n²+2²)+.+n²/(n²+n²)]}
=lim(n->∞){(1/n)[1/(1+(1/n)²)+1/(1+(2/n)²)+.+1/(1+(n/n)²)]}
=∫(0,1)dx/(1+x²) (由定积分定义得)
=(arctanx)|(0,1)
=π/4-0
=π/4