What is the canonical way to understand such things as
$$(1+q + q^2 + \ldots q^9)(1+q^{10} + q^{20} + \ldots q^{90})((1+q^{100} + q^{200} + \ldots q^{900}) \ldots \quad ?$$
It is a problem from a combinatorics course which I don't attend but rather solve problems from there sometimes.
So I know how to compute it but I don't like that no one explained what does it even mean. I can make sense of it in the following way. Define $\mathbb{K}[[x]]$ as the limit if the diagram $$\mathbb{K}[x]/(q) \leftarrow \mathbb{K}[x]/(q^2) \leftarrow \ldots$$
in the category of, say, topological algebras (multi-normed) (each $\mathbb{K}[x]/(q^i)$ is finite dimensional vector space so it has just one norm and we put the limit topology on $\mathbb{K}[[x]]$.
In this topology the product in question converges to the correct answer.
What else can be done to understand problems like this?