已知无穷等比数列{an}首项为1,公比为q,前n项和为Sn,求lim(Sn/Sn+1)

问题描述:

已知无穷等比数列{an}首项为1,公比为q,前n项和为Sn,求lim(Sn/Sn+1)

(1)当q=1时,Sn=n;S(n+1)=(n+1)
lim(Sn/Sn+1)=n/(n+1)=1
(2)当q=-1时,n为偶数Sn=0;S(n+1)=1,极限=0
n为奇数,Sn=1;S(n+1)=0,极限不存在;
(3)当q≠±1时:
Sn=a1·(1-q^n)/(1-q)
S(n+1)=a1·[1-q^(n+1)]/(1-q)
Sn/S(n+1)=(1-q^n)/[1-q^(n+1)]
∴lim(Sn/Sn+1)=(1-q^n)/[1-q^(n+1)]
若|q|>1:
lim(Sn/Sn+1)=lim(1/q^n -1)/[1/q^n-q]=(-1)/(-q)=1/q
若|q|