Is every p-adic extension a finite-tailed laurent series? At the bottom of page 2 of this: < Lagarias states that the p-adic expansion of any rational number $x$ is a finite-tailed Laurent series in powers of $p$. But I appear to be misunderstanding something since I make the expansions of fractions infinite. Take $\frac{2}{3}$ for example: $$-1=\overline 1_2$$ $$\frac{-1}{3}=\overline{01}_2$$ $$\frac{2}{3}=1-\frac{1}{3}=\overline{01}10_2$$ Which would make its expansion $2^1+2^2+2^4+2^6+2^8...$ But that is not what I would call a finite-tailed expansion, since it is clearly infinite! Am I missing something? Or by finite-tailed, does he simply mean it has no negative powers of $p$?

By finite tailed, he means that the index of the sum formally defining the $p$-adic expansion has a lower limit:

$$ \frac{a}{b} = \sum_{k\geq n_0} a_kp^k $$

For the rationals, your speculation about finite-tailed meaning no negative powers is "right," in the sense that no negative powers does imply finite tailed. But you could also start from, say, $p^{-4}$ and still be finite-tailed.