Write $P(x; a) = \sum_{n \geq 0} a_n x^n$. Is there a way to show that there exists a choice of nonzero sequence $a_n$ for which $P(x_k; a) = 0$ for $k = 1, \dots, n$, and $x_1, \dots, x_n \in \mathbf{R}$?
More generally, suppose $X \subset \mathbf{R}$, when is
$$\{ a \in \mathbf{R}^\mathbf{N} \setminus \{0\} \mid P(x; a) = 0, \mbox{for all}~x \in X \} $$ nonempty? (I know that when $X$ contains a neighborhood of 0.)
For your first question, consider the polynomial $$ f(x) = \prod_{i = 1}^n (x - x_i) $$ This polynomial will have a finite power series expansion $$ f(x) = \sum_{i = 1}^n a_n x^n $$ which, because it is finite, will converge everywhere. Note that this is nothing but the statement that $f$ is a polynomial in $x$ of degree $n$.
As to your second question, it is a fact from complex analysis that if $X \subseteq \mathbb{R}$ has a limit point in $\mathbb{R}$, then the power series must be identically $0$. In other words, the set, as you've defined it, is empty.
As far as I know, when $X$ doesn't have a limit point, anything can happen.
