Estimate of Taylor coefficients

Question

I have a real function f(x) different from zero and not proportional to $\exp(x)$ analytic at origin, $$ f(x)=\sum_{n=0}^{+\infty}c_n x^n,\ \forall x:|x|<\delta $$ To derive a result, I should require that $\forall n\geq N$ $$|c_n|\leq \frac{A}{n!} $$ with $A$ being a constant $\geq 0.$ Which hypothesis should I give to $f(x)$ to have that $|c_n|\leq \frac{A}{n!}$?
Thank you in advance.

Answer 1

Note that since $c_n=\frac{f^{(n)}(0)}{n!}$, if you require that there exists some constant $A$ such that for any $n$, $|f^{(n)}(0)|\leq A$, then your condition can be obtained.