We set, for all $t \in I$, $$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$ Show that $\forall t \in I , g ( t ) ^ { 2 } = 2 g ( t ) - 4 t$.
We set, for all $t \in I$,
$$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$
Show that $\forall t \in I , g ( t ) ^ { 2 } = 2 g ( t ) - 4 t$.