For $\alpha \in \mathbb{R}$, recall, without giving a proof, the power series expansion of $( 1 + x ) ^ { \alpha }$ on $]-1,1[$.
Justify the formula:
$$\forall x \in ]-1,1[ , \quad \frac { 1 } { \sqrt { 1 - x } } = \sum _ { n = 0 } ^ { + \infty } \frac { \binom { 2 n } { n } } { 4 ^ { n } } x ^ { n }$$