Order of a formal series, congruence modulo $x^N$, and the $x$-adic notions of convergence and Cauchy sequence

For a nonzero $f\in R\mathrm{⟦}x\mathrm{⟧}$, its formal order is

${\mathrm{ord}}_x(f):=\min \{n\in \mathbb{N}:[x^n]f\mathrm{\ne }0\},$

and ${\mathrm{ord}}_x(0):=+\mathrm{\infty }$. We use the conventions $m<+\mathrm{\infty }$, $m+(+\mathrm{\infty })=+\mathrm{\infty }$, and $\min (m,+\mathrm{\infty })=m$.

For $N\in \mathbb{N}$, write

$f\equiv g(\mathrm{m}\mathrm{o}\mathrm{d}{x}^N)$

when $[x^n]f=[x^n]g$ for every $n<N$. Equivalently, ${\mathrm{ord}}_x(f-g)\ge N$. At $N=0$ the coefficient condition is empty, so any two series are congruent modulo $x^0=1$.

A sequence $(f_j{)}_{j\ge 0}$ converges $x$-adically to $f$ if for every $N$ there is $J$ such that $f_j\equiv f(\mathrm{m}\mathrm{o}\mathrm{d}{x}^N)$ whenever $j\ge J$. It is $x$-adically Cauchy if for every $N$ there is $J$ such that $f_j\equiv f_k(\mathrm{m}\mathrm{o}\mathrm{d}{x}^N)$ whenever $j,k\ge J$. Thus convergence and the Cauchy condition mean eventual stability of each finite coefficient prefix; they do not assert analytic convergence.