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The title of the thread doesn't adequately describe the question I want to ask, so here it is:

Suppose we have two infinite series, [itex]\sum_{n=1}^{\infty}a_n[/itex] and [itex]\sum_{n=1}^{\infty}b_n[/itex], both of which are convergent. Also suppose [itex]a_n \leq b_n[/itex] for all [itex]n[/itex], and [itex]a_n < b_n[/itex] for at least one [itex]n[/itex]. Is it possible that they both converge to the same limit? Or can we say that [itex]\sum_{n=1}^{\infty}a_n[/itex] is strictly smaller than [itex]\sum_{n=1}^{\infty}b_n[/itex]?

If the answer to the above question is that they can have the same limit, then what if [itex]a_n<b_n[/itex] for infinitely many [itex]n[/itex]?

Thanks!

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# Can two different infinite series converge to the same limit?

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