I discuss (with references, but without proof) the Levy-Steinitz Theorem in Section 2 of the following document:
http://www.math.uga.edu/~pete/UGAVIGRE08.pdf
In particular, the version I give describes precisely the set of limits of convergent rearrangements in terms of the subspace of directions of absolute convergence of the series. As a special case, if no one-dimensional projection is absolutely convergent, then indeed one can rearrange the series to converge to any vector in $\mathbb{R}^n$.
