Neumann series in Banach algebras

The Neumann series can be generalized to a Banach algebra $𝒜$ with identity element $e$.Thus we have:

If $x\in 𝒜$ is such that then $e-x$ is invertible with inverse given by

$${(e-x)}^{-1}=\sum _{n=0}^{\mathrm{\infty }}{x}^n$$

and

$$\parallel {(e-x)}^{-1}\parallel \le \frac{1}{1-\parallel x\parallel }$$