Is the set of analytic functions equal to the set of polynomials?

Question

Taylor's theorem and the Taylor series give us an equivalence between analytic functions and polynomials. For example, $C^\omega \ni e^x = \sum_{n=1}^\infty \frac{x^n}{n!} \in \mathbb{P}$, where $C^\omega$ is the set of all complex analytic functions and $\mathbb{P}$ is the set of all complex polynomials. Since, by definition, an analytic function has a polynomial (Taylor series) which converges to the function, does that imply that $C^\omega = \mathbb{P}$? Or is it not correct to say that $\sum_{n=0}^\infty a_nx^n \in \mathbb{P}$?