lim(n/(n^2+1)+n/(n^2+2^2)+……+n/(n^2+n^2))n^2指n的平方

Question

lim(n/(n^2+1)+n/(n^2+2^2)+……+n/(n^2+n^2))
n^2 指n的平方

Answer 1

lim(n/(n^2+1)+n/(n^2+2^2)+……+n/(n^2+n^2))
=lim1/n*[1/(1+1/n^2)+1/(1+(2/n)^2)+……+1/(1+(n/n)^2)]
根据定积分的定义.
对于∫1/(1+x^2) x∈（0,1）
=lim∑1/n*(1/(1+(i/n)^2)
=lim1/n*[1/(1+1/n^2)+1/(1+(2/n)^2)+……+1/(1+(n/n)^2)]
则lim(n/(n^2+1)+n/(n^2+2^2)+……+n/(n^2+n^2))
=lim1/n*[1/(1+1/n^2)+1/(1+(2/n)^2)+……+1/(1+(n/n)^2)]
=∫1/(1+x^2) x∈（0,1）
=arctanx x∈（0,1）
=π/4