Let $X : \Omega \rightarrow \mathbb { R }$ be a real-valued random variable. We assume that $X ( \Omega )$ is countable and we use the notation from question 2: $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ with $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. We also assume that, for all integer $n \in \mathbb { N } , X$ admits a moment of order $n$ and that there exists a real $R > 0$ such that $$\mathbb { E } \left( | X | ^ { n } \right) = O \left( \frac { n ^ { n } } { R ^ { n } } \right) \quad \text { when } n \rightarrow + \infty$$ Show that for all $n \in \mathbb { N }$ and all $y \in \mathbb { R } , \left| \mathrm { e } ^ { \mathrm { i } y } - \sum _ { k = 0 } ^ { n } \frac { ( \mathrm { i } y ) ^ { k } } { k ! } \right| \leqslant \frac { | y | ^ { n + 1 } } { ( n + 1 ) ! }$.
Let $X : \Omega \rightarrow \mathbb { R }$ be a real-valued random variable. We assume that $X ( \Omega )$ is countable and we use the notation from question 2: $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ with $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$.\\
We also assume that, for all integer $n \in \mathbb { N } , X$ admits a moment of order $n$ and that there exists a real $R > 0$ such that
$$\mathbb { E } \left( | X | ^ { n } \right) = O \left( \frac { n ^ { n } } { R ^ { n } } \right) \quad \text { when } n \rightarrow + \infty$$
Show that for all $n \in \mathbb { N }$ and all $y \in \mathbb { R } , \left| \mathrm { e } ^ { \mathrm { i } y } - \sum _ { k = 0 } ^ { n } \frac { ( \mathrm { i } y ) ^ { k } } { k ! } \right| \leqslant \frac { | y | ^ { n + 1 } } { ( n + 1 ) ! }$.