I really like to think of $\mathbb{N}$-graded $R$-modules as power series $\bigoplus_{n \in \mathbb{N}} M_n \otimes X^{\otimes n}$, where each "coefficient" $M_n$ is an $R$-module and $X$ is the graded module concentrated degree $1$ and which is $R$ there. Therefore, we have a category of power series, where a morphism is just a family of morphisms between the coefficients. Actually this is a symmetric monoidal category - the tensor product is given by some convolution. And the same works if we replace $\mathsf{Mod}(R)$ by any cocomplete symmetric monoidal category. This is spelled out for example in Section 5.4 of my thesis.
In order to get a connection to power series in analysis, we might endow the unit interval $[0,1]$ with the structure of a cocomplete symmetric monoidal category (cf. Example 3.1.6 in loc.cit.): We use the usual ordering to make it a (thin) category and the usual multiplication to make it a symmetric monoidal category. Colimits are given by suprema. Therefore, we get a cocomplete symmetric monoidal category of sequences valued in $[0,1]$. This is again just an order with a multiplication, where we have $(a_n) \leq (b_n)$ iff $a_n \leq b_n$ for all $n$, and $((a_p) \cdot (b_q))_n = \sup_{p+q=n} a_p \cdot b_q$.
Also notice that coends in category theory capture some ideas of (definite) integration. See MO/78471 for some intuition.
