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 $$r _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N + 1 } e ^ { - 1 / x }, \quad R _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t$$ Show that, if $N \in \mathbb { N } ^ { * }$ and $x > 0$, $$\left| R _ { N } ( x ) \right| \leqslant \left| r _ { N } ( x ) \right|$$ Deduce that $R _ { N + 1 } ( x ) = o \left( r _ { N } ( x ) \right)$ when $x \rightarrow 0$.
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
$$r _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N + 1 } e ^ { - 1 / x }, \quad R _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t$$
Show that, if $N \in \mathbb { N } ^ { * }$ and $x > 0$,
$$\left| R _ { N } ( x ) \right| \leqslant \left| r _ { N } ( x ) \right|$$
Deduce that $R _ { N + 1 } ( x ) = o \left( r _ { N } ( x ) \right)$ when $x \rightarrow 0$.