Question

lim(n→无穷)(1/根号下n^2+1^2+1/根号下n^2+2^2+…+1/根号下n^2+n

2

Answer 1

原式 = 1/n * (n/sqrt(n^2+1) + ... + n/sqrt(n^2+k^2) + ... + n/sqrt(n^2+n^2))
= 1/n * (1/sqrt(1+1/n^2) + ... + 1/sqrt(1+k^2/n^2) + ... + 1/sqrt(1+n^2/n^2))
令f(x)=1/sqrt(1+x^2)，那么上式就是f(x)从x=0到1的定积分
f(x)的原函数F(x)=arctanx
原式 = arctan1 - arctan0 = pi/4