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 $\left| a _ { n + 1 } \right| \geqslant \frac { \left| a _ { n } \right| } { 2 ( n + 1 ) }$ for all $n \geqslant q$. Deduce that, for all integer $n \geqslant q , \left| a _ { n } \right| \geqslant \frac { q ! \left| a _ { q } \right| } { 2 ^ { n - q } n ! }$.