The set of all limits of sub-series of an absolute convergent series

The set is a compact subset of $\mathbb{C}$

Let $K=\{0,1\}^{\mathbb{N}^*}$ and endow $K$ with the product topology where each $\{0,1\}$ has the discrete topology. It can be shown (even without AC/Tychonoff's Theorem) that $K$ is compact. For $i\in\mathbb{N}^*$ we denote by $p_i:K\to\{0,1\}$ the canonical projection on the $i$-th component. We recall that the topology on $K$ is the least topology that makes each $p_i$ continuous.

Let $(a_i)_{i\geq1}$ be a sequence of real (or complex) numbers such that the series $\sum_i a_i$ is absolutely convergent. Define the mapping $$\begin{array}[t]{ccccc}\mathbf{a}&:&K&\longrightarrow&\mathbb{C}\\&&(c_i)_{i\geq1}&\longmapsto&\displaystyle\sum_{i=1}^{+\infty}a_i c_i.\end{array}$$ Since we assumed that the series $\sum_i a_i$ is absolutely convergent, it is not hard to check that the mapping $\mathbf{a}$ is well-defined (e.g., using the comparison test).

We now show that $\mathbf{a}$ is continuous: let $(c_i)_{i\geq1}\in K$ and let $\varepsilon>0$. Since the series $\sum_i a_i$ is absolutely convergent, there exists $N\in\mathbb{N}^*$ such that $\sum_{i=N+1}^{+\infty}\lvert a_i\rvert<\varepsilon$. Consider the open subset $$O_N=\bigcap_{i=1}^N p_i^{-1}\bigl(\{c_i\}\bigr)$$ of $K$. By definition of the product topology, $O_N$ is an open subset of $K$ (indeed, $O_N$ is a finite intersection of inverse images of open sets by the continuous mappings $p_i$). Let $(d_i)_{i\geq1}\in O_N$. By definition, this means that $(d_i)_{i\geq1}$ is a sequence with values in $\{0,1\}$ and $$\forall i\in\{1,\ldots,N\},\ d_i=c_i.$$ Then $$\Bigl\lvert\mathbf{a}\bigl((d_i)_{i\geq1}\bigr)-\mathbf{a}\bigl((c_i)_{i\geq1}\bigr)\Bigr\rvert=\left\lvert\sum_{i=N+1}^{+\infty}(d_i-c_i)a_i\right\rvert\leq\sum_{i=N+1}^{+\infty}\lvert a_i\rvert<\varepsilon.$$ Hence $\mathbf{a}$ is continuous.

It is a standard fact that the image of a compact space by a continuous mapping is compact: we hence conclude that $\mathbf{a}(K)$ is a compact subset of $\mathbb{C}$.

Necessary condition for the set to be connected

Let us denote the set of all possible limits of "sub-series" by $M$. gniourf_gniourf has proven that $M$ is always a compact subset of $\mathbb{C}$. Since you also asked for connectedness:

Claim: If $M$ is connected, then the sequence $(a_i)_{i\in\mathbb{N}}$ does not become constant, or if $M = \{0\}$.

Proof: If the sequence $(a_i)$ becomes constant (i. e. $\exists\ N$ such that $a_i = 0$ for all $i > N$), then $M$ contains only finitely many values (at most $2^N$). Hence $M$ is disconnected, unless $M$ has only one element.

I feel very much like this should be an equivalence-statement, but I couldn't find a proof for the other direction so far.