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 $n \geq 1$. Show that if $f$ is a function $E$, then the numerical series $\sum_{n=0}^{\infty} \frac{b_n}{n!} \alpha^n$ converges for every real number $\alpha$. We denote its value by $f(\alpha)$.
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 $n \geq 1$.
Show that if $f$ is a function $E$, then the numerical series $\sum_{n=0}^{\infty} \frac{b_n}{n!} \alpha^n$ converges for every real number $\alpha$. We denote its value by $f(\alpha)$.