Answer to Question #159887 in Calculus for sai

Question #159887

Suppose limn→∞ (Sn − 1)/ (Sn + 1) = 0. Prove that limn→∞ Sn = 1.


1
Expert's answer
2021-02-02T04:45:07-0500

Given that limnSn1Sn+1=0lim_{n \to \infty} \frac {S_n-1}{S_n+1}=0

    limn(Sn1)limn(Sn+1)=0\implies \frac {lim_{n \to \infty} (S_n-1)}{lim_{n \to \infty}( S_n+1)}=0 ........(1)

Now from (1) we can say that as the exists , therefore limn(Sn+1)0lim_{n \to \infty}( S_n+1) \neq 0

So the only possibility is that , limn(Sn1)=0lim_{n \to \infty} (S_n-1)=0

    limnSn=1\implies lim_{n \to \infty} S_n=1

Which completes the proof.


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