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

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


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

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

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I’m not sure what you mean. If $S$ exists, $\lim_{n\to\infty}a_n=0$, but $S$ can be arbitrarily large. For instance, $\sum_{i=1}^\infty 2^{n-i}=2^n$. – Brian M. Scott Oct 2 '11 at 1:36
Oh, that answered my question, thx! – ablmf Oct 2 '11 at 1:41
up vote 1 down vote accepted

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. – anon Oct 2 '11 at 1:58

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