Grandi's series appears to have different values depending on how you bracket the terms, e.g. $(1 - 1) + (1 - 1) + (1 - 1) + ... = 0$, while $1 + (-1 + 1) + (-1 + 1) + ... = 1$. This isn't a problem today because we define the sum of a series as the limit of its partial sums (which is this case just doesn't exist). And the observation is that you can't perform rearrangements or even regroupings of a series' terms (unless it is absolutely convergent).
I often see p-adic series being manipulated in similar ways, e.g. equation 2.1 here, where: $\overline{n_0n_1 \dots n_{k-1}} = 1(n_0n_1 \dots n_{k-1}) + p^k (n_0n_1 \dots n_{k-1}) + p^{2k}(n_0n_1 \dots n_{k-1}) + \dots = (n_0n_1 \dots n_{k-1}) ( 1 + p^k + p^{2k} + \dots)$.
I've been trying to convince myself this is safe/meaningful. Looking at partial sums (or really, the values in $\varprojlim \mathbb Z/ p^n \mathbb Z$) I can convince myself the left side equals the right side. But I'm not sure how to formally interpret the middle as a p-adic number (other than just rearranging it to be either the left or right side). I know of p-adic numbers as either formal infinite series of the form $\sum_v a_v p^v$ or elements in $\varprojlim \mathbb Z/ p^n \mathbb Z$. (Or as cauchy sequences).
Should I just think of the middle step as being for intuition?