Let $\forall n=0,1,2,\dots$, $\alpha_{n}(x)$ are POLYNOMIALS in $x$. Next, let for all $x\neq0$ the power series $$\sum_{n=0}^{\infty}\alpha_{n}(x)t^{n}$$ has positive radius of convergence. Can one say the radius of convergence of the series $$\sum_{n=0}^{\infty}\alpha_{n}(0)t^{n}$$ is also positive?

My quess is NO, but I have no example.