First, I noticed that the nth derivative of $f$ at $0$ is $n!\cdot a_n$, but this does not really help me to construct a sequence of numbers that can not be generated by the sequence ${f(0),f'（0），f''(0),\ldots}$. How should I construct the sequence?
Thank you in advance.
Find a sequence that grows fast enough to make the radius of convergence of $$ \sum_{n=0}^\infty \frac{a_n}{n!} z^n $$ less than $1$.