Let $(A,||\cdot||)$ be a commutative Banach algebra over $\mathbb{C}$. Consider a formal power series $f(z):=\sum_{n=0}a_n z^n\in A[[z]]$ and let $$ r:=\frac{1}{\limsup\limits_{n\to\infty}||a_n||^{1/n}} $$ For $b\in A$ denote by $\rho(b)$ its spectral radius.
Is it generally true that $f(b)$ is divergent (in the $||\cdot||$-topology) for any $b\in A$ with $\rho(b)>r$?
I believe one can easily show in certain cases that $f(b)$ is divergent if $$ \rho(b)> \frac{1}{\limsup\limits_{n\to\infty}\rho(a_n)^{1/n}}=:R, $$ but $R$ can be much bigger than $r$.
No, and more generally you can make no statement along these lines that depends only on $\rho(b)$. For instance, let $a,b\in A$ be elements with nonzero spectral radius such that $ab=0$. Let $a_n=c_na$ for some scalars $c_n$. Then $r$ can be made arbitrary by choosing $c_n$ appropriately, but $f(b)$ will always converge since all terms after the first term will be $0$.