You wrote a series. If you mean that it converges for all $x\in R^n$ than this is an entire function that is the series also converges for $x\in C^n$. An entire algebraic function is a polynomial, indeed. This follows from the growth estimate and Cauchy's inequalities, for example.

I only do not understand why did you add $C^\infty$ condition. The sum of a power series is of course $C^\infty$.

You wrote a series. If you mean that it converges for all $x\in R^n$ than this is an entire function that is the series also converges for $x\in C^n$. An entire algebraic function is a polynomial, indeed. This follows from the growth estimate and Cauchy's inequalities, for example.