I am finding some behaviour I didn't entirely expect in a system of mine based on rather simple arithmetico-geometric progression (but with a little twist).
To begin with I considered $h_{n+1} = h_n + \frac{Rh_n-E}{c}$ and $X_n=Rh_{n-1}$. We may assume that $R$ and $c$ are always constant.
Formulae can be obtained for $h_n$ and $X_n$ as arithmetico-geometric progressions, assuming constant $E$. It can subsequently be shown that:
$$\frac{X_n-E}{X_n(E=0)} = 1-\frac{E}{Rh_0}$$
This again assumes, of course, constant $E$. The behaviour I would like to investigate assumes not constant $E$ but rather discretely-changeable $E$.
The question arises because I observed a certain particular case. Namely, when $h_0=12, R=190*10^6, c=75*10^6$, and with $E=3.65*10^8$ for n=1, $E=10^9$ for n=2, $E=3.65*10^9$ for n=3, $E=10^{10}$ for n=4, $E=3.65*10^{10}$ for n=5, and $E=10^{11}$ for n=6 (and the same for n>6), then I observe that
$$lim_{n \to \infty}{(\frac{X_n-E}{X_n(E=0)})} \approx 0.47970946$$
My question is, is there any way to derive the value of this limit? Or in general, if given $h_0, R, c$ as well as $E_1$ (the value of $E$ for $n=1...n_1$), $E_2$ (the value of $E$ for $n=n_1+1...n_2$), ..., $E_j$ for $n=n_{j-1}+1...\infty$, a formula that can give the value of this limit, without having to just crunch the numbers recursively?