We are given a probability space $( \Omega , \mathcal { A } , \mathbb { P } )$. We set for every integer $n \geqslant 0$, $B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$ where $S(n,k)$ is the number of partitions of $\llbracket 1,n \rrbracket$ into $k$ parts. Let $m$ be a strictly positive integer, and $\mathbb { E } \left( Y ^ { m } \right) = \sum _ { n = 0 } ^ { \infty } n ^ { m } \mathbb { P } ( Y = n )$. We assume in this question that $Y$ follows the Poisson distribution with parameter 1. IV.C.1) Show that for all $n \in \mathbb { N } , B _ { n } = \mathbb { E } \left( Y ^ { n } \right)$. IV.C.2) Deduce that for every polynomial $Q ( X )$ with integer coefficients, the series $\sum _ { n = 0 } ^ { + \infty } \frac { Q ( n ) } { n ! }$ is convergent and its sum is of the form $N e$, where $N$ is an integer.
We are given a probability space $( \Omega , \mathcal { A } , \mathbb { P } )$. We set for every integer $n \geqslant 0$, $B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$ where $S(n,k)$ is the number of partitions of $\llbracket 1,n \rrbracket$ into $k$ parts. Let $m$ be a strictly positive integer, and $\mathbb { E } \left( Y ^ { m } \right) = \sum _ { n = 0 } ^ { \infty } n ^ { m } \mathbb { P } ( Y = n )$.
We assume in this question that $Y$ follows the Poisson distribution with parameter 1.
IV.C.1) Show that for all $n \in \mathbb { N } , B _ { n } = \mathbb { E } \left( Y ^ { n } \right)$.
IV.C.2) Deduce that for every polynomial $Q ( X )$ with integer coefficients, the series $\sum _ { n = 0 } ^ { + \infty } \frac { Q ( n ) } { n ! }$ is convergent and its sum is of the form $N e$, where $N$ is an integer.