In the real numbers, under the standard topology, there exist sequences $ \{x_n\} $, with $x_n \neq 0 $ such that the infinite sum
$$ \sum_{n=0}^{\infty} x_n $$
converges.
However, in the integers, under addition, no such series exists. Perhaps the series $x_{2n} = 1$, $x_{2n + 1} = -1$, could be considered convergent, but this feels like it depends a lot on the topology, and I feel like there should be some natural sense in which it is not considered convergent (I guess when viewed as a subspace/group of the real numbers).
Is there a theory which answers the question: Which groups have non-trivial convergent series? Is this even an interesting/well formed question?