We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by
$$F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$$
Show that $F$ is well defined and of class $\mathscr { C } ^ { \infty }$ on $] 0 , + \infty [$.