User:Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 7

Theorem
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.

Then:
 * $1 + p^k + p^{2k} + p^{3k} + \ldots = \dfrac 1 {1 - p^k}$

Proof
Let $S_n$ be the partial sum:
 * $\ds S_n = \sum_{j = 0}^n p^{j k}$

We have: