Justify the existence of a sequence of real numbers $\left( a _ { n } \right) _ { n \in \mathbb { N } }$ such that
$$\forall x \in ] - 1,1 \left[ , \quad \sqrt { 1 + x } = 1 + \sum _ { n = 0 } ^ { + \infty } a _ { n } x ^ { n + 1 } , \right.$$
and, for all $n \in \mathbb { N }$, express $a _ { n }$ using a binomial coefficient.