I have a problem where I am able to find to solution using an iterative process. The problem (simplified) goes as follow:
$$ x_n=x_{n-1}^2\cdot K $$
Where $ K<1 $. Actually $ K \approx0.01685$
I know the value of $x_0$ ($x_0=3.089$), so using a software like excel or matlab I can iterate that problem and I see that the value converge quite quickly towards 0. The information I want is the sum of the elements, which I believe, should converge to a constant value since each elements converges towards 0.
$$ \sum_{n=0}^{\infty}x_n$$
I would like try solving the problem using an analytical approach to find the limit of the convergence as a function of $x_0$ and $K$.
I tried to solve this using the Z-Transform. But have some troubles.
First, I am not sure if I should write : $X=X^2\cdot Z^{-1}\cdot K$ or $X=(X\cdot Z^{-1})^2\cdot K$
The second solution doesn't seems right as it doubles the delay, but I feel like a skipped a step if I use the first version.
If I use the first version of the equation. I then substitute $Z=e^{sT}$ where I set $T=1$
I then have $$ X=X^2\cdot Z^{-1}\cdot K$$ $$ X = X^2\cdot e^{-s}\cdot K $$ $$ X = \frac{1}{Ke^{-s}}$$
And from there, I am not sure how to handle the rest. If I can make the inverse Laplace, I hope to get a function of a variable that represent the iteration and that will surely converge to the value I am looking for.
Any idea? I am not sure if I used a correct approach or not. Any help would be appreciated!
Thank you
By a change of variable $y=Kx$, you can temporarily ignore $K$, as
$$y_n=y_{n-1}^2.$$
Then by induction, the general term of the series is
$$y_n=y_0^{2^n}$$
and AFAIK, there is no closed form for
$$\sum_{k=0}^n y_0^{2^k}.$$
For small $y_0$ (yours is about $0.052$), the approximation
$$y_0+y_0^2+y_0^4+y_0^8$$ is excellent.