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