实数a1,a2,…ak.他们之和为0.求证极限：lim{a1*(n+1)^（1/2）+a2*(n+2)^(1/2)+…ak*(n+k)^(1/2)}=0

题目

实数a1,a2,…ak.他们之和为0.求证极限：lim{a1*(n+1)^（1/2）+a2*(n+2)^(1/2)+…ak*(n+k)^(1/2)}=0
a1*(n+1)^（1/2)表示a1乘以n+1的1/2次方

答案

=lim[a1*(n+1)^（1/2）+a2*(n+2)^(1/2)+…+a(k-1)*(n+k-1)^(1/2)-(a1+a2+…+a(k-1))*(n+k)^(1/2)]
=lim{a1*[(n+1)^（1/2）-(n+k)^(1/2)]+a2*[(n+2)^(1/2)-(n+k)^(1/2)]+…a(k-1)*[(n+k-1)^(1/2)-(n+k)^(1/2)]}
=lim{a1*[(n+1)^（1/2）-(n+k)^(1/2)]}+lim{a2*[(n+2)^(1/2)-(n+k)^(1/2)]}+…+lim{a(k-1)*[(n+k-1)^(1/2)-(n+k)^(1/2)]}
=0+0+...+0(k个0,即可数个0)
=0