For $p \notin \mathbb{N}^*$, $p \neq 0$, let $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ be a non-zero power series solution of $(E_p)$, with $q$ a natural integer such that $q > p$ and $\left| a _ { n } \right| \geqslant \frac { q ! \left| a _ { q } \right| } { 2 ^ { n - q } n ! }$ for all $n \geqslant q$. Show that the function $\psi : \left\lvert\, \begin{array} { c c c } \mathbb { R } _ { + } ^ { * } & \rightarrow & \mathbb { R } \\ x & \mapsto & \sum _ { n = q } ^ { + \infty } \left| a _ { n } \right| x ^ { n } \end{array} \right.$ is not an element of $E$.
For $p \notin \mathbb{N}^*$, $p \neq 0$, let $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ be a non-zero power series solution of $(E_p)$, with $q$ a natural integer such that $q > p$ and $\left| a _ { n } \right| \geqslant \frac { q ! \left| a _ { q } \right| } { 2 ^ { n - q } n ! }$ for all $n \geqslant q$. Show that the function $\psi : \left\lvert\, \begin{array} { c c c } \mathbb { R } _ { + } ^ { * } & \rightarrow & \mathbb { R } \\ x & \mapsto & \sum _ { n = q } ^ { + \infty } \left| a _ { n } \right| x ^ { n } \end{array} \right.$ is not an element of $E$.