# Does this sum have an upper bound?

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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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
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, $$\sum_{i=1}^\infty2^{n-i}=2^n.$$
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