We consider the functions $F$ and $G$ defined by the formulas $$\begin{aligned}
& \forall x \in ]-1,1[ , \quad F ( x ) = \sum _ { n = 0 } ^ { + \infty } P \left( S _ { n } = 0 _ { d } \right) x ^ { n } \\
& \forall x \in [ - 1,1 ] , \quad G ( x ) = \sum _ { n = 1 } ^ { + \infty } P ( R = n ) x ^ { n }
\end{aligned}$$ Show that $$\forall x \in ]-1,1[ , \quad F ( x ) = 1 + F ( x ) G ( x ) .$$ Determine the limit of $F ( x )$ as $x$ tends to $1^{-}$, discussing according to the value of $P ( R \neq + \infty )$.
We consider the functions $F$ and $G$ defined by the formulas
$$\begin{aligned}
& \forall x \in ]-1,1[ , \quad F ( x ) = \sum _ { n = 0 } ^ { + \infty } P \left( S _ { n } = 0 _ { d } \right) x ^ { n } \\
& \forall x \in [ - 1,1 ] , \quad G ( x ) = \sum _ { n = 1 } ^ { + \infty } P ( R = n ) x ^ { n }
\end{aligned}$$
Show that
$$\forall x \in ]-1,1[ , \quad F ( x ) = 1 + F ( x ) G ( x ) .$$
Determine the limit of $F ( x )$ as $x$ tends to $1^{-}$, discussing according to the value of $P ( R \neq + \infty )$.