题目

lim(n→∞)[1/(3n+1)+1/(3n+2)+~1/(3n+n)]

提问时间：2021-01-08

答案

lim(n→∞) 1/(3n + 1) + 1/(3n + 2) + ...+ 1/(3n + n)
= lim(n→∞) 1/[n(3 + 1/n)] + 1/[n(3 + 2/n)] + ...+ 1/[n(3 + n/n)]
= lim(n→∞) (1/n)[1/(3 + 1/n) + 1/(3 + 2/n) + ...+ 1/(3 + n/n)]
= ∫(0→1) dx/(3 + x)
= ∫(0→1) d(3 + x)/(3 + x)
= [ln(3 + x)]：(0→1)
= ln(3 + 1) - ln(3 + 0)
= ln4 - ln3
= ln(4/3)