  Let $\sum_{n=1}^\infty x_n$ be a conditionally convergent series in a Banach
space and let $\tau$ be a permutation of natural numbers. We study the set
$\operatorname{LIM}(\sum_{n=1}^\infty x_{\tau(n)})$ of all limit points of a
sequence $(\sum_{n=1}^p x_{\tau(n)})_{p=1}^\infty$ of partial sums of a
rearranged series $\sum_{n=1}^\infty x_{\tau(n)}$. We give full
characterization of limit sets in finite dimensional spaces. Namely, a limit
set in $\mathbb{R}^m$ is either compact and connected or it is closed and all
its connected components are unbounded. On the other hand each set of one of
these types is a limit set of some rearranged conditionally convergent series.
Moreover, this characterization does not hold in infinite dimensional spaces.
  We show that if $\sum_{n=1}^\infty x_n$ has the Rearrangement Property and
$A$ is a closed subset of the closure of the $\sum_{n=1}^\infty x_n$ sum range
and it is $\varepsilon$-chainable for every $\varepsilon>0$, then there is a
permutation $\tau$ such that $A=\operatorname{LIM}(\sum_{n=1}^\infty
x_{\tau(n)})$. As a byproduct of this observation we obtain that series having
the Rearrangement Property have closed sum ranges.
