P-adic sequence with a limit outside rational numbers

Question

I'm reviewing my $p$-adic notes and had stumbled upon the following sequence of rational numbers: $$x_n = \sum_{k = 0}^n p^{k^2}.$$

It's obviously a fundamental sequence (edit: with respect to $p$-adic norm, not the classical one!). If I showed that it does not possess a rational limit, I would have a witness of $\mathbb Q$'s incompleteness. How to proceed?

There is a related sequence converging to $\frac 1 {1-p}$: $$y_n = \sum_{k = 0}^n p^{k}.$$

Answer 1

For nothing but the fact of the incompleteneless of $\Bbb Q$, why not just note that for $p\ne2$, $\sqrt{1+p}\in\Bbb Z_p$, by Hensel, and if $p>3$, it’s not rational. As a matter of fact, $\sqrt{1+p}$ has a perfectly good $p$-adically convergent series from the expansion that comes from Binomial Theorem.