Given an abelian group G and a non-trivial sequence in G, when will it be possible
to construct a Hausdroff topology on G that allows the sequence to converge? As one might expect
of such a naive question, the answer is far too complicated for a simple response. The purpose of
this paper is to provide some insights to this question, especially for the integers, the rationals, and
any abelian groups containing these groups as subgroups. We show that the sequence of squares
in the integers cannot converge to 0 in any Hausdroff group topology. We demonstrate that any
sequence in the rationals that satisfies a sparseness condition will converge to 0 in uncountably
many different Hausdorff group topologies.