  This article considers the numerical inversion of the power series
$p(x)=1+b_{1}x+b_{2}x^{2}+\cdots$ to compute the inverse series $q(x)$
satisfying $p(x)q(x)=1$. Numerical inversion is a special case of triangular
back-substitution, which has been known for its beguiling numerical stability
since the classic work of Wilkinson (1961). We prove the numerical stability of
inversion of power series and obtain bounds on numerical error. A range of
examples show these bounds to be quite good. When $p(x)$ is a polynomial and
$x=a$ is a root with $p(a)=0$, we show that root deflation via the simple
division $p(x)/(x-a)$ can trigger instabilities relevant to polynomial root
finding and computation of finite-difference weights. When $p(x)$ is a
polynomial, the accuracy of the computed inverse $q(x)$ is connected to the
pseudozeros of $p(x)$.
