Proposition 12

If the series whose general term is $z^n$ is convergent in $A$, then $e - z$ is a unit of $A$ and we have

Conversely, if $||z|| < 1$ and if $e - z$ is a unit in $A$, then the series whose general term is $z^n$ is convergent and formula (7) is valid.

For each $p > 0$ we have

If the series whose general term is $z^n$ is convergent and if $y$ is its sum, then $z^n$ tends to 0 as $n \to + \infty$; hence by passing to the limit in (8) we have $(e - z)y = e$; similarly we prove that $y(e - z) = e$, and hence $y = (e - z)^{-1}$ (note that this part of the argument is valid in any topological ring which has an identity element).

Conversely, if $||z|| < 1$, then since $||z^{p+1}|| \leq ||z||^{p+1}$, it follows that $z^{p+1}$ tends to 0 as $p \to +\infty$; multiplying both sides (8) on the left by $(e-z)^{-1}$ and letting $p$ tend to infinity, we see that the series whose general term is $z^n$ converges and has $(e-z)^{-1}$ as its sum.