Could be the radius of convergence $0$ for EGF if $b_n > n!$ for $n>n_{0}$?

Question

let $f (x)$ be exponential Generating function (EGF) has radius of convergence $r$ defined as:$f(x)= \sum_{n=0}^{+\infty}b_{n}\frac{x^n}{n!}$ such that $b_n > n!$ for $n>n_{0}$

Could be the radius of convergence $0$ for exponential Generating function (EGF) if $b_n > n!$ for $n>n_{0}$ ?.