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Let $\sum_{n=1}^\infty x_n$ be a conditionally convergent series in a Banachspace 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 asequence $(\sum_{n=1}^p x_{\tau(n)})_{p=1}^\infty$ of partial sums of arearranged series $\sum_{n=1}^\infty x_{\tau(n)}$. We give fullcharacterization of limit sets in finite dimensional spaces. Namely, a limitset in $\mathbb{R}^m$ is either compact and connected or it is closed and allits connected components are unbounded. On the other hand each set of one ofthese 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 rangeand it is $\varepsilon$-chainable for every $\varepsilon>0$, then there is apermutation $\tau$ such that $A=\operatorname{LIM}(\sum_{n=1}^\inftyx_{\tau(n)})$. As a byproduct of this observation we obtain that series havingthe Rearrangement Property have closed sum ranges.

On the set of limit points of conditionally convergent series