Question

求数列极限,高手帮帮忙吧

求：数列极限：[1/√(2+n^2)]+[1/√(1+2+n^2)+....+[1/(1+n+n^2)]
求：数列极限：[(1/n^2+1)]+[]1/(n^2+2)]+....+[1/(n^2+n)]
两题多要解法

Answer 1

(1)设A=[1/√(2+n^2)]+[1/√(1+2+n^2)+....+[1/(1+n+n^2)]
则A>=[1/√(1+n+n^2)]+[1/√(1+n+n^2)+....+[1/(1+n+n^2)]=n/√(1+n+n^2)
A<=[1/√(2+n^2)]+[1/√(2+n^2)+....+[1/(2+n^2)]=n/√(2+n^2)
lim n/√(1+n+n^2)=lim1/√(1/n^2+1/n+1)=1
lim n/√(2+n^2)=lim1/√(2/n^2+1)=1,因此limA=1
(2)B=[(1/n^2+1)]+[1/(n^2+2)]+....+[1/(n^2+n)]
B>=[(1/n^2+n)]+[1/(n^2+n)]+....+[1/(n^2+n)]=n/(n^2+n)
B<=[(1/n^2+1)]+[1/(n^2+1)]+....+[1/(n^2+1)]=n/(n^2+1)
limn/(n^2+n)=0
limn/(n^2+1)=0
因此limB=0