Let $f(x) = \sum_{n=0}^{\infty} c_n x^n \in \mathbf{Q}\llbracket x \rrbracket$ be a power series whose coefficients $c_n$ are integers. Show that if there exists a real number $\alpha \geq 1$ such that the numerical series $\sum_{n=0}^{\infty} c_n \alpha^n$ converges, then $f$ is a polynomial.