I'll give you an example in $\mathbb{C}[[x, y]]$:
$$\sum_{n,m\in \mathbb{N}} \binom{n+m}{m} x^ny^m =\sum_{k\in \mathbb{N}}\sum_{n+m=k} \binom{n+m}{m}x^ny^m =\sum_{k\in\mathbb{N}} (x+y)^k=\frac{1}{1-x-y}$$
Why are we allowed to write the first equality? How is that rigorously defined in our space? We have infinite sum of vectors in $\mathbb{C}[[x, y]]$.
$R[[X]]$ is not just a vector space, it is a topological vector space. There are several ways you can introduce topology (as written here), but the important thing is that it is the finest topology such that all sequences of partial sums $$S_n = a_0+a_1X+\ldots+a_nX^n$$ converge to the corresponding power series $\displaystyle\sum_{i=0}^\infty a_iX^i$.
This is precisely why we are able to write $$(1,0,0,\ldots)+(0,1,0,\ldots)+\ldots = (1,1,1,\ldots)$$ since we think of the LHS as a sequence of partial sums that converges to the RHS.