Let $\displaystyle Q_p[x] = \left\{\frac{p(x)}{q(x)} \mid \, p(x),q(x) \in \mathbb{F}_p[x], \, q(x) \neq 0 \right\}$ denote the field of fractions extending $\mathbb{F}_p[x]$. If we consider the sequence $\displaystyle \left\{\sum_{k=0}^n x^i\right\}$, that, in $\mathbb{R}$, would converge to $\displaystyle \frac{1}{1-x}$ with the usual metric $| \cdot|$ kept in mind, then, say, over elements in $\mathbb{F}_2$, what metric can we construct such that we make sense of the "intuitive limit" (or what should feel like) $$\sum_{k=0}^\infty x^k = \frac{1}{1+x} \, ?$$ Moreover, over $\mathbb{F}_p$, what would the sum $\displaystyle \sum_{k=0}^\infty x^k$ converge to?

If this has been done so before, references would very much be appreciated. In regards to what I have done, I have entertained the use of the Hamming metric, but its application only makes sense in $\mathbb{F}_p[x]$ and I'm forced to construct an extension of the Hamming metric for $Q_p[x]$, which I have not done so successfully (or at least acceptably so). Any suggestions?

Let $\displaystyle Q_p[x] = \left\{\frac{p(x)}{q(x)} \mid \, p(x),q(x) \in \mathbb{F}_p[x], \, q(x) \neq 0 \right\}$ denote the field of fractions extending $\mathbb{F}_p[x]$. If we consider the sequence $\displaystyle \left\{\sum_{k=0}^n x^i\right\}$, that, in $\mathbb{R}$, would converge to $\displaystyle \frac{1}{1-x}$ with the usual metric $| \cdot|$ kept in mind, then, say, over elements in $\mathbb{F}_2$, what metric can we construct such that we make sense of the "intuitive limit" (or what should feel like) $$\sum_{k=0}^\infty x^k = \frac{1}{1+x} \, ?$$

If this has been done so before, references would very much be appreciated. In regards to what I have done, I have entertained the use of the Hamming metric, but its application only makes sense in $\mathbb{F}_p[x]$ and I'm forced to construct an extension of the Hamming metric for $Q_p[x]$, which I have not done so successfully (or at least acceptably so). Any suggestions?