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For a formal power series $f(x)={\sum }_{n\ge 0}{a}_n{x}^n$ over a commutative ring $R$, a reciprocal is a series $g$ with $fg=1$. It exists exactly when $a_0$ is a unit. The constant coefficient forces $b_0=a_0^{-1}$, and the higher coefficients are determined successively byThus the reciprocal is unique and uses no analytic convergence assumption. For example, $(1-x{)}^{-1}=1+x+x^2+\cdots$ in $R[[x]]$. Newton inversion of a formal power series accelerates the computation when many coefficients are needed.