Exercise 4, § 3

Let $(\alpha_n)_{n\geqslant 0}$ be a sequence in $\mathbf{R}_+^*$ such that $\alpha_0= 1,\alpha_{m+n}\leqslant \alpha_m\alpha_n$ for all integers $m$ and $n$, and such that $\alpha^{1/n}_n\rightarrow 0$. Let A be the set of formal series

$x=\sum_{n=0}^{\infty}x_nz^n\in \mathbf{C}[[z]]$ such that

Prove that A is a commutative unital Banach algebra topologically generated by $z$, and that the unique maximal ideal of A is the set of $x\in A$ without constant term. (Observe that $z$ is quasi-nilpotent.)