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If we have an infinite sequence of positive numbers whose sum is

$ S = \sum_{i=1}^\infty a_n $

and

$ \lim_{n \to \infty} a_n = 0 $

Can we draw conclusion that $S$ has an constant upper bound?

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    Oh, that answered my question, thx!2011-10-02

1 Answers 1

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Turning the comment into an answer:

If $S$ exists, it’s necessarily true that $\lim\limits_{n\to\infty}a_n=0$, but $S$ can still be arbitrarily large. For instance, $\sum_{i=1}^\infty2^{n-i}=2^n.$

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    Not only that, but $S$ might not exist even given $\lim\limits_{n\to\infty}a_n=0$, e.g. the harmonic series.2011-10-02