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)
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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)
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)
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|
当q=1时Sn=n,即Sn/Sn+1=n/n+1, lim(Sn/Sn+1)=1;
当q≠1时,Sn=1*(1-q^n)/(1-q)=(1-q^n)/(1-q);
即Sn/Sn+1=(1-q^n+1)/(1-q^n);
此时分两种情况讨论:
① 0即lim(Sn/Sn+1)=lim「(1-q...
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当q=1时Sn=n,即Sn/Sn+1=n/n+1, lim(Sn/Sn+1)=1;
当q≠1时,Sn=1*(1-q^n)/(1-q)=(1-q^n)/(1-q);
即Sn/Sn+1=(1-q^n+1)/(1-q^n);
此时分两种情况讨论:
① 0即lim(Sn/Sn+1)=lim「(1-q^n+1)/(1-q^n)」=(1-0)/(1-0)=1;
② q>1时,lim(q^n)趋向于无穷大,
Sn/Sn+1=(1-q^n+1)/(1-q^n)=(q^n+1 - 1)/(q^n-1);
相对于q^n,1可以忽略不计,
即limSn/Sn+1=lim「(q^n+1 - 1)/(q^n-1)」=(q^n+1)/q^n=q;
综上所述:当0〈 q ≤1时,lim(Sn/Sn+1)=1;
当q 〉1时,limSn/Sn+1=q 。
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