Composition $f\circ g$ of formal series when the outer series is a polynomial or the inner series has zero constant term

For $f={\sum }_{n\ge 0}{a}_n{x}^n$ and $g\in R\mathrm{⟦}x\mathrm{⟧}$, define the formal composition

$f\circ g:={\sum }_{n\ge 0}{a}_n{g}^n$

in either of these cases:

1. $f$ is a polynomial, so the sum is finite; or
2. $[x^0]g=0$, so ${\mathrm{ord}}_x(g^n)\ge n$ and the displayed family is summable.

Both rules give the same result when both apply. In particular $f\circ 0=[x^0]f$, $f\circ x=f$, and $x\circ g=g$ whenever the displayed compositions are admissible.

If $f$ has infinitely many nonzero coefficients and $[x^0]g\mathrm{\ne }0$, the expression is not defined over a bare commutative ring: even its constant coefficient could require an infinite sum in $R$. This is a failure of formal local finiteness, not a question of analytic convergence.