Suppose $A$ and $B$ are some constant ($A,B\in\mathbb{R}$)
Is there a simple expression for $x$, where $x$ is:
$$ x=A+B[A+B[A+B[\cdots]]]] $$ "..." indicates the pattern repeats forever.
In other words, it is recursive: $$ x=A+Bx $$
It is very tempting to just do: $$ x=\frac{A}{1-B} $$
However, is this correct? I don't think is that simple.
For context, I come from an engineering background and my math isn't as rigorous. Please enlighten me. Does this wiki resembles this kind of series?
Edit 1: I don't think the expression for $x$ is simple, because: $$ x\neq \frac{A}{1-B} $$
Given recurrence relation
$$x_{n+1}=A+Bx_{n},$$
induction shows
$$x_{n}=\begin{cases} B^{n}x_{0}+A\frac{1-B^{n}}{1-B} & B\neq1\\ x_{0}+nA & B=1 \end{cases}.$$
If $|B|<1,$ then $$\lim_{n\to \infty}x_n=\frac{A}{1-B}.$$