Series related to the Mandelbrot set Apologies if this question has been asked here before. The Mandelbrot set is the set of complex numbers $c$ for which the iteration $z_{n+1}(c)=z_n (c)^2+c$ with $z_0(c)=0$ does not diverge (we write $z_\infty(c)=\lim\limits_{n\rightarrow\infty}z_n(c)$ where it converges). Thus every $z_n(c)$ can be written as a polynomial in $c$ of order $2^{n-1}$, so we define:
$$z_n(c)=\sum_{j=0}^\infty \beta_{n,j}c^j$$
where $\beta_{n.j}$ is only nonzero for $j\le2^{n-1}$. Using the Mandelbrot iteration formula and the Cauchy product, we get:
$$\beta_{n+1,j}=\begin{cases}\sum\limits_{k=0}^j\beta_{n,k}\beta_{n,j-k}&j\ne1\\2\beta_{n,0}\beta_{n,1}+1&j=1\end{cases}\tag{1}$$
Now $\beta_{n,0}=0\;\;\forall n\ge0$ and $\beta_{n,1}=1\;\;\forall n\ge1$ and using $(1)$ we can show that $\beta_{n+1,2}=\beta_{n,1}^2=1\;\;\forall n\ge1$, that $\beta_{n+1,3}=2\;\;\forall n\ge2$ and in general that $\beta_{n,j}=\beta_j\;\;\forall n\ge j$ where:

$$\beta_j=\begin{cases}0&j=0\\1&j=1\\\sum\limits_{k=0}^j \beta_k\beta_{j-k}&j\ge2\end{cases}$$

(I calculated the first few terms $0,1,1,2,5,14,42,132,429,...$ by hand, but the OEIS would not give me a general formula). Thus the first $n+1$ terms of $z_n(c)$ are $\beta_i c^i$ as seen here in red:
$$z_0(c)=\color{#ff0000}{0}$$
$$z_1(c)=\color{#ff0000}{0+c}$$
$$z_2(c)=\color{#ff0000}{0+c+c^2}$$
$$z_3(c)=\color{#ff0000}{0+c+c^2+2c^3}+\color{#0000ff}{c^4}$$
$$z_4(c)=\color{#ff0000}{0+c+c^2+2c^3+5c^4}+\color{#0000ff}{6c^5+6c^6+4c^7+c^8}$$
and so on. Thus as $n\rightarrow\infty$ the first $n+1$ terms of $z_n(c)$ will tend to the following series:

$$z(c)=\sum_{j=0}^\infty\beta_j c^j\tag{2}$$

However, because of the extra $2^{n-1}-n$ terms in $z_n(c)$ (shown in blue), I assume that $\{z_n(c)\}$ will not always converge to $z(c)$, especially since the Mandelbrot set is not a disc (and power series converge within discs), but I do not know of any way of describing the extra terms which will enable a connection to be made between the convergence of the Mandelbrot iteration and the convergence of $(2)$.
My questions are the following: 


*

*For what $c$ are $\color{blue}{the\ blue\ terms}$ small enough that $z(c)=z_\infty(c)$, 

*How if at all is the convergence of $(2)$ connected with the convergence of the iteration $z_{n+1}=z_n^2+c$? 

*I would also to like to know whether there is a closed form for $z(c)$ [see comment]  

*What is the radius of convergence of $(2)$?

 A: *

*Despite expanding finite iterations, there's a series maps the exterior of a unit disk to the exterior of the $M$ set.
\begin{align*}
   \psi(z) &= z+\sum_{m=0}^{\infty} \frac{\beta_{0,m+1}}{z^m} \\
   z_{n} &= \sum_{m=0}^{\infty} \beta_{n,m} z^{2^n-m} \\
   &= z^{2^{n}}+o(1)
 \end{align*}

See John H. Ewing, Glenn Schober, The area of the Mandelbrot Set

*
Strictly speaking, we won't say the iteration converges when the final states are oscillating. Hence, not all the points on the Mandelbrot set converge to a limit.  All points in $n$-periodic cycles $(n>1)$ or chaotic bands are bounded, so they belong to the Mandelbrot set but do not converge.


*If you insist for how interior of $M$ is mapped to its final states.  Please see derivation about period $1$ cycle and its final state below:
Snapshot I, snapshot II and snapshot III
In conclusion,
$$c=\frac{re^{i\phi}}{2}-\frac{r^2e^{2i\phi}}{4} \mapsto 
   z=\frac{re^{i\phi}}{2}$$ where $(r,\phi) \in [0,1] \times [0,2\pi)$

*For period one cycle, $z$ and $c$ can be expressed in quadratic:
$$z^2-z+c=0$$
Taking the branch enclosing the super-attractive point $c=0$,
\begin{align}
    z &= \frac{1-\sqrt{1-4c}}{2} \\
    &= \sum_{k=1}^{\infty} \frac{(-1)^{n-1}c^{n}}{2^{2n-1}}
       \binom{\frac{1}{2}}{n} \\
    &= c+c^2+2c^4+5c^6+14c^8+42c^{10}+\ldots
  \end{align}
which has the radius convergence of $\dfrac{1}{4}$.
The boundary of the mapping:
$$c=\frac{e^{i\theta}}{4} \mapsto
   z=\frac{1-\sqrt{|\sin \frac{\theta}{2}|+|\sin \frac{\theta}{2}|^2}}{2}-
     \frac{i\operatorname{sgn} (\sin \theta) \sqrt{|\sin \frac{\theta}{2}|-|\sin \frac{\theta}{2}|^2}}{2}$$

*The iteration and series agree only when $|c| \le \dfrac{1}{4}$.

Below is the comparison between the exact iterations (blue) and the series summation (red):



The green boundaries refer pre-image and image enclosed for period two cycle.



*Animation of successive iterations of the Mandelbrot Set:

