Let $A$ be a domain and $X \subset A$. Then $(X) = \{a_1x_1 + \cdots + a_nx_n \mid n \in \mathbb{N},\, a_i \in A,\, x_i \in X\}$
What I understand from the above theorem is that if $x\in X$ then there is a way to express it as a linear combination of a finite number of elements of $X$. If $X$ were an infinite set, whose elements are $x_i$ for all $i\in \mathbb{N}$, then there exists an element of $(X)$ that can be expressed as a sum of infinite quantity of $x_i$?
No, the linear combination must be finite. Infinite sums might not even be defined.