Question

怎么用极限严格定义求数列n\（a^n）的极限

Answer 1

a^n=(1+(a-1))^n
=n + n·(a-1) + n(n-1)/2·(a-1)² + .
=n·[1 + (a-1) + (n-1)/2·(a-1)² + .]
n/a^n
=1/[1 + (a-1) + (n-1)/2·(a-1)² + .]
≤1/[(n-1)/2·(a-1)²]
=2/[(n-1)·(a-1)²]
对任意的ε>0
取N=不小于2/(ε·(a-1)²) + 1的整数

Answer 2

lim n/a^n=lim 1/(a^n·ln a)=0（洛必达法则）
a^n=[(a-1)+1]^n>1+n(a-1)+n(n-1)/2·(a-1)^2=1+t
n/a^n<n/(1+t)<ε
1/ε<1/n+t/n
当n>1时，1/n+t/n<1+t/n
1/ε<1+t/n
2/[ε(a-1)^2]<2/[(a-1)^2]+2/(a-1)+n-1
n>[2/ε+a^2-4a+1]/[(a-1)^2]=s
N=max{1,[s]}
当n>N时，有|n/a^n-0|<ε
所以lim n/a^n=0