Can every power series be represented as a Taylor series?
More concrete: Given an arbitrary power series $\sum_{n=0}^\infty a_n (x-x_0)^n$, is there always a $C^\infty$-function $f$ such that $\sum_{n=0}^\infty a_n (x-x_0)^n = \sum_{n=0}^\infty \frac{f^{n}(x_0)}{n!} (x-x_0)^n$, i.e. for each sequence $(a_n)_{n\in\mathbb N}$ exists a $C^\infty$-function $f$ with $a_n = \frac{f^{n}(x_0)}{n!}$?
To further expand the comment of @James,
given an arbitrary sequence of numbers $a_n$ and fixed point $x_0$, there exists a $C^\infty_c(\Bbb R)$ function $\phi$ such that $\phi^{(n)}(x_0)=a_n$ (this fact is useful in the theory of distributions).