证明:若lim(n→∞)an=a,则lim(n→∞)(a1+2a2+…+nan)/n^2=a/2

Question

如题所述

Answer 1

(a1+2a2+…+nan)/n^2 = (1/n)∑<i=1→n>ai·(i/n)
当n→∞时，可将1/n看作dx，i/n看作x，范围(积分限)为从0到1
则原式变为
(1/n)∑<i=1→n>ai·(i/n)
=∫<0→1>x·a dx
=a·∫<0→1>x dx
=a/2