Theorem. Let be a commutative ring with non-zero unity. A formal power series
is invertible in the ring iff is invertible in the ring .
Proof. . Let have the multiplicative inverse . Since
we see that , i.e. is an invertible element (unit) of .
. Assume conversely that is invertible in . For making from a formal power series
the inverse of , we first choose . For all already defined coefficients let the next coefficient be defined as
This equation means that
vanishes for all ; since , the product of the formal power series (1) and (2) becomes simply equal to 1. Accordingly, is invertible.