令S(n)=1+(1+2)+(1+2+3)+...+(1+2+...+n)
设an = 1+2+...+n = n(n+1)/2
则
Sn = a1 + a2 + ...+ an
= 1(1+1)/2 + 2(2+1)/2 + ...+ n(n+1)/2
= [1(1+1) + 2(2+1) + ...+ n(n+1)]/2
= [(1*1 + 2*2 + ...+ n*n) + (1 + 2 + ...+ n)]/2
= [bn + an]/2
其中bn = 1*1 + 2*2 + ...+ n*n
= n(n+1)(2n+1)/6
故
Sn = [bn + an]/2
= [n(n+1)(2n+1)/6 + n(n+1)/2]/2
lim(1/n^3+(1+2)/n^3+...+(1+2+...+n)/n^3)=1/6