TITLE: Complex infinite sum convergence problem
QUESTION [5 upvotes]: Suppose that the complex infinite sum $ \sum_{n=1}^{\infty}(-1)^{n}Z_n$ converges. 
Define $A \subset \mathbb{C}$ by
$A=\{{z\in\mathbb{C}\mid\exists f:\mathbb{N}\underset{on}{\overset{1-1}{\rightarrow}}\mathbb{N},z= \sum_{n=1}^{\infty}Z_{f(n)}}\}$
Prove that $A=\mathbb{C}$ or that  $\exists a, b \in \mathbb{C}$ such that $A = \{a+tb \mid t\in \mathbb{R}\}$.
--edit-- in bounty message I mean solution of the problem.

REPLY [4 votes]: Let $\displaystyle \sum_{n \geq 1} X_n$ be a conditionally convergent series in the $d$-dimensional real Euclidean space $\mathbb{R}^{d}$. Then the set $Y$ of all possible sums of rearrangments of this series is an affine subspace of $\mathbb{R}^{d}$. When $d=1$, this is the well know Riemann rearrangement theorem and the proof is not very difficult.
Addendum: A good reference on this is the following: 
The Remarkable Theorem of Levy and Steinitz,
Peter Rosenthal,
The American Mathematical Monthly,
Vol. 94, No. 4 (Apr., 1987), pp. 342-351
I also realized that you do not need conditional convergence because of the way you stated your conclusion (b = 0 covers absolutely convergent case). So I removed that trivial remark.