 Inverse of a formal power series

Here an inverse of a formal power series means its multiplicative inverse: $f(x{)}^{-1}f(x)=1$. The formal power series reciprocal exists exactly when the constant coefficient of $f$ is a unit in the coefficient ring. Thus $1-x$ is invertible in $\mathbb{Z}[[x]]$, while $2+x$ is not, though it is invertible in $\mathbb{Q}[[x]]$.

A compositional inverse solves $f(g(x))=x$, which is a different problem. For instance, the multiplicative inverse of $1-x$ is $1+x+x^2+\cdots$, whereas the compositional inverse of $x+x^2$ starts $x-x^2+2x^3-\cdots$. Stating which operation is inverted prevents a common ambiguity.