I have a question about limits at infinity, particularly, about a limit I have seen in the context of infinite series convergence.(adsbygoogle = window.adsbygoogle || []).push({});

Let's say we have an infinite series where the the sequence of partial sums is given by {S(n)} and also, it is convergent and the sum is equal to S. Then we know that the limit of S(n) as n approaches infinity is S, but from this, can we also say the the limit of S(n+1) is equal to S?

Well, based from the textbook I was reading, it is. I actually have an intuitive idea of why, but I'd rather see something more of a 'formal' explanation, probably something which has epsilons. I tried verifying with an epsilon-N proof but I'm not that confident.

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# Limit on infinity

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