 Reciprocal of a power series

If $f(x)={\sum }_{n\ge 0}{a}_n{x}^n$ has $a_0\ne 0$, its reciprocal has a unique formal power series $1/f(x)={\sum }_{n\ge 0}{b}_n{x}^n$. Equating coefficients in the Cauchy product $f\cdot (1/f)=1$ gives the displayed recurrence relation. In particular,For a convergent power series, continuity and $f(0)\ne 0$ give a neighborhood without zeros, where the reciprocal is analytic and this series converges. Its radius need not equal the original radius: zeros of $f$ can introduce closer reciprocal singularities.