求n项和数列极限,通式为i/(i+n)就是n->无穷,1/(n+1)+2/(n+2)+3/(n+3)+...+n/2n;

Question

求n项和数列极限,通式为i/(i+n)
就是n->无穷,1/(n+1)+2/(n+2)+3/(n+3)+...+n/2n;

Answer 1

你好,这个极限不收敛的,就是极限是∞.
这个可以根据定积分的定义来说明.
原极限=lim(n->∞) ∑ [i/(n+i)]=lim(n->∞) n (1/n)∑(i/n)/[1+(i/n)]
=lim n * {lim (1/n)∑(i/n)/[1+(i/n)]}
=lim n * ∫(0->1) [x/(1+x)]dx
=(1-ln2) *lim n
=∞