Sequence of Ratios

Question

Let $\{a_n\}_{n\ge 0}$ be a positive real sequence and define $$r_n=\frac{a_{n+1}}{a_n},\quad n\ge 0$$ Suppose that we know the formal power series of $a_n$, i.e. we know the following: $$A(x)=\sum_{n\ge 0}a_nx^n$$ Then, is there any method by which we can find the formal power series of $r_n$, i.e. can we find the following as a function of $A(x)$, without explicitly putting values for $r_n$? $$R(x):=\sum_{n\ge 0}r_nx^n$$ At least, can we find $\displaystyle \lim_{n\rightarrow \infty}r_n$(if it exists) in terms of $A(x)$? Thank you.