Question

数列｛an｝是等比数列,首项不为1,公比q,且Tn=Sn/Sn-1,求lim(n→∞)Tn

Answer 1

q=1时,

Sn=nA1

Tn=Sn/S(n-1)=(nA1)/((n-1)A1)=n/(n-1)=1/(1-1/n)

lim(n→∞)Tn=1

q≠1时,

Sn=A1(1-q^n)/(1-q)

Tn=(1-q^n)/(1-q^(n-1))

q>1时,

Tn=(1/q^n-1)/(1/q^n-1/q)

n→∞

1/q^n→0

lim(n→∞)Tn=(0-1)/(0-1/q)=q

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