Unit criterion for noncommutative formal power series ($F\text{ is a unit}⟺F_{\varnothing }\ne 0$)

A noncommutative formal power series over a field is a unit exactly when its constant term is nonzero. Necessity follows because the constant term is a ring homomorphism to the coefficient field. For sufficiency, write $F=c(1+h)$ with $c\ne 0$ and $h$ of zero constant term. The formal geometric series ${\sum }_{j\ge 0}(-h{)}^j$ is well-defined because only $j\le d$ can contribute to a word of length $d$. Its left and right products with $1+h$ are $1$, by the finite geometric-series identity in each fixed degree. Multiplying by $c^{-1}$ gives the inverse of $F$.