We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$.
For $N \in \mathbb { N } ^ { * }$ and $x > 0$, we set
$$r _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N + 1 } e ^ { - 1 / x }$$
Show that, for $0 < x < 1 / 2$, the sequence $\left( \left| r _ { N } ( x ) \right| \right) _ { N \geqslant 1 }$ is decreasing up to a certain rank, then increasing.