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 $$R _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t$$ Specify the domain of convergence of the series $\sum _ { k \geqslant 1 } ( - 1 ) ^ { k - 1 } ( k - 1 ) ! x ^ { k }$ and show that the sequence $\left( R _ { N } ( x ) \right) _ { N \geqslant 1 }$ is not bounded.
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
$$R _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t$$
Specify the domain of convergence of the series $\sum _ { k \geqslant 1 } ( - 1 ) ^ { k - 1 } ( k - 1 ) ! x ^ { k }$ and show that the sequence $\left( R _ { N } ( x ) \right) _ { N \geqslant 1 }$ is not bounded.