Consider the ring of formal power series $R[[x]]$ and given $\sum a_{n}x^{n}$ define a metric on $R[[x]]$ as follows: $d((a_n),(b_n))=2^{-k}$ where $k$ is the smallest natural number such that $a_{k} \neq b_{k}$ (if no such $k$ exists we define their distance to be zero).
Let $\{x_{i}: i \in \mathbb{N}\}$ be a sequence of elements of $R[[x]]$ (i.e each $x_{i}$ is a formal power series). Now for each natural number $j$ and each sequence $\{x_{i}: i \in \mathbb{N}\}$ let $G(x_{i},j)$ be the set of all natural numbers i such that $x_{i}(j) \neq 0$.
For example $x_{1}$ is a formal series, and $x_{1}(j)$ means the $j$-th coefficient of the formal series $x_{1}$.
From now on assume $G(x_{i},j)$ is finite.
Define a formal series $\sum_{i=0}^{\infty} x_{i}$ as follows:
$(\sum_{i=0}^{\infty} x_{i})(j):= \sum_{i \in G(x,j)} x_{i}(j)$, i.e the $j$-th coefficient of $\sum_{i=0}^{\infty} x_{i}$ is equal to $\sum_{i \in G(x,j)} x_{i}(j)$.
Question: let $\{x_{i}: i \in \mathbb{N}\}$ be a sequence of formal power series and let $S_{n}=\sum_{i=0}^{n} x_{i}$. Is it true that the sequence $\{S_{n}: n \in \mathbb{N}\}$ converges as $n \rightarrow \infty$ (with respect the metric defined previously) to $\sum_{i=0}^{\infty} x_{i}$?