Good morning everyone, I'd like to discuss with the following exercise and one fact of theory that I'm not understanding.
Let's take the following Recurrence relation :
$\begin{cases} & \quad x_{n+1} = (x_{n})^{n}\\ & \quad x_{1}= \frac{1}{2} \end{cases}$
I'd like to know when I can and should discretize the Recurrence relation to obtain $f(x) = x^{n}$ and when I shouldn't.
Because if I discretize I obtain that $f(0)=0,f(1)=1$ so $0,1$ are fixed points.
After that, I'd like to study the function (limits,growth,ecc) which results are different if I explicit (or at least maybe that's the error) the Recurrence in the following way :
$(\frac{1}{2})^{(n-1)!}$
Same with this :
$x_{n+1}= x_{n} \cdot arctan(n)$ , in which I think I found that the explicit value of $x_{n} = \frac{1}{2015} \prod\limits_{i=1}^{n} arctan(i)$, in conflict (I think) the discretization.
Your help would be appreciated to clarify my doubts,
Thanks anyway.