P-adic analysis

Remarks

The absolute value $\abs{\cdot}_p$ satisfies the strong triangle inequality $\abs{x + y}_p \le \max(\abs{x}_p, \abs{y}_p)$. Hence a series $\sum_n a_n$ in $\Q_p$ converges if and only if $a_n \to 0$.

Example: $\abs{p^n}_p = p^{-n} \to 0$, so $\sum_{n \ge 0} p^n$ converges in $\Q_p$, and its sum is $\frac{1}{1 - p}$, since $(1 - p) \sum_{n=0}^{N} p^n = 1 - p^{N+1} \to 1$.

Non-example: the same series $\sum_{n \ge 0} p^n$ diverges for the usual absolute value on $\R$; its study is part of real analysis, not of $p$-adic analysis.

Two typical facts. The ring $\Z_p$ of $p$-adic integers is compact and totally disconnected. The exponential series $\sum_n x^n / n!$ converges in $\Q_p$ exactly when $\abs{x}_p < p^{-1/(p-1)}$.