A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$. Let $f$ be a function $E$ that is not a polynomial. Show that there exists $R > 0$ such that the numerical series $\sum_{n=0}^{\infty} b_n \alpha^n$ diverges for every real number $\alpha$ with $|\alpha| > R$.
A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying:
(a) $f$ is a solution of a differential equation;
(b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Let $f$ be a function $E$ that is not a polynomial. Show that there exists $R > 0$ such that the numerical series $\sum_{n=0}^{\infty} b_n \alpha^n$ diverges for every real number $\alpha$ with $|\alpha| > R$.