Proposition 2

Let A be a Banach algebra and $x$ an element of A. The series $\sum_{n=1}^{\infty}\lambda^nx^n$, considered as a power series in $\lambda$, has radius of convergence $\varrho (x)^{-1}$. If A is unital and if $\varrho (x)<1$, then $1-x$ is invertible and has as inverse $\sum_{n=0}^{\infty}x^n$.

The series $\sum^{\infty}_{n=1}\lambda^nx^n$ has radius of convergence

(lim sup$_{n\rightarrow+\infty}\|x^n\|^{1/n}$)$^{-1}=\varrho (x)^{-1}$

(cf. VAR, R1, p. 23, 3.1.4). Suppose that A admits a unit element. If $\varrho (x)<1$, the series $\sum_{n=0}^{\infty}x^n$ is therefore absolutely convergent. Since

for every integer $k\geqslant 0$, the element $1-x$ is invertible and its inverse is equal to $\sum_{n=0}^{\infty}x^n$.