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:

$1 + p^k + p^{2 k} + p^{3 k} + \cdots = \dfrac 1 {1 - p^k}$

Proof

Let $S_n$ be the partial sum:

$\ds S_n = \sum_{j \mathop = 0}^n p^{j k}$

We have:

\(\ds \paren {1 - p^k} S_n\)

\(=\)

\(\ds \paren {1 - p^k} \sum_{j \mathop = 0}^n p^{j k}\)

\(\ds \)

\(=\)

\(\ds \paren {\sum_{j \mathop = 0}^n p^{j k} } - p^k \paren {\sum_{j \mathop = 0}^n p^{j k} }\)

distributing the partial sum across $1 - p^k$

\(\ds \)

\(=\)

\(\ds \paren {\sum_{j \mathop = 0}^n p^{j k} } - \paren {\sum_{j \mathop = 0}^n p^{\paren {j + 1} k} }\)

distributing $p^k$ across the partial sum

\(\ds \)

\(=\)

\(\ds 1 - p^k + p^k - p^{2 k} + p^{2 k} - \ldots - p^{n k} + p^{n k} - p^{\paren {n + 1} k}\)

rearranging terms

\(\ds \)

\(=\)

\(\ds 1 - p^ {\paren {n + 1} k}\)

as the series telescopes

\(\ds \leadsto \ \ \)

\(\ds \norm {\paren {1 - p^k} S_n - 1}_p\)

\(=\)

\(\ds \norm {1 - p^{\paren {n + 1} k} - 1}_p\)

\(\ds \)

\(=\)

\(\ds \norm {- p^{\paren {n + 1} k} }_p\)

cancelling

\(\ds \)

\(=\)

\(\ds \norm {p^{\paren {n + 1} k} }_p\)

Norm of Negative in Division Ring

\(\ds \)

\(=\)

\(\ds \dfrac 1 {p^{\paren {n + 1} k} }\)

Definition of P-adic Norm on Rational Numbers

\(\ds \)

\(\to\)

\(\ds 0\)

as $n \to \infty$

\(\ds \leadsto \ \ \)

\(\ds \lim_{n \mathop \to \infty} \paren{1 - p^k} S_n\)

\(=\)

\(\ds 1\)

Definition of Convergent P-adic Sequence

\(\ds \leadsto \ \ \)

\(\ds \lim_{n \mathop \to \infty} S_n\)

\(=\)

\(\ds \dfrac 1 {\paren {1 - p^k} }\)

Multiple Rule for Sequences in Normed Division Ring

\(\ds \leadsto \ \ \)

\(\ds \sum_{j \mathop = 0}^\infty p^{j k}\)

\(=\)

\(\ds \dfrac 1 {\paren {1 - p^k} }\)

Definition of Series

$\blacksquare$