Is there a $C^{\infty}$ function $f$ such that the radius of convergence of $\sum\_{n=0}^\infty \frac{f^{(n)}(x\_0)}{n!} (x-x\_0)^n$ is $0$?

By the way, I know the following fact: 

If $f(x) = \sum\_{n=0}^\infty a\_n x^n$ for $x \in (-R, R)$ for a positive real number $R$, then $f(x) = \sum\_{n=0}^\infty \frac{f^{(n)}(0)}{n!} x^n$ 
 for $x \in (-R, R)$. 

So, if there is a $C^{\infty}$ function $f$ such that the radius of convergence of $\sum\_{n=0}^\infty \frac{f^{(n)}(0)}{n!} x^n$ is $0$, then we cannot write $f(x) = \sum\_{n=0}^\infty a\_n x^n$ for $x \in (-R, R)$ for a positive real number $R$.