We set, for all $t \in I$,
$$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$
Deduce that there exists a function $\varepsilon : I \rightarrow \{ - 1,1 \}$ such that
$$\forall t \in I , \quad g ( t ) = 1 + \varepsilon ( t ) \sqrt { 1 - 4 t } .$$