Related to that previous question I have another still open detail problem.
Consider the sequence of evaluations at some given $x$ $$ \small \begin{array} {} z_0 &=& e^x \\ z_1 &=& e^{2x} & + (-1x) e^x \\ z_2 &=& e^{3x} & + (-2x) e^{2x} & + (-1x)^2 e^x/2! \\ z_3 &=& e^{4x} & + (-3x) e^{3x} & + (-2x)^2 e^{2x}/2! & + (-1x)^3 e^x/3! \\ \cdots &=& \cdots \\ z_n &=& e^{(n+1)x} & +(-nx) e^{nx} & \ldots & + (-2x)^{n-1} e^{2x}/(n-1)! & + (-1x)^n e^x/n! \\ \end{array}$$
The quotient $ \displaystyle q_n(x) = {z_{n+1} \over z_n }$ seems to approximate some constant value (of course depending on $x$), so the sequence of $z_n$ approaches asymptotically a geometric series. I couldn't relate this quotient to $x$ by trying with empirical data (except where $x=1$, then apparently $\lim_{n \to \infty} q_n(x) = 1$ and asymptotically with increasing $x$ approximates $e^x$ ). So my question is:
Q: Is there an exact closed-form expression for $q_n(x)$ as $n \to \infty$ ?
[update]
Here is some example data: $$ \small { \begin{array} {r|r|} x & q=q_{200}(x) & \log(q) & x - \log(q) \\ \hline \\ 1 & 1.00000000000 & -2.430 E-86 & 1.00000000000 \\ 2 & 4.92155363457 & 1.59362426004 & 0.406375739960 \\ 3 & 16.8010161907 & 2.82143937212 & 0.178560627878 \\ 4 & 50.4352530011 & 3.92069039487 & 0.0793096051271 \\ 5 & 143.324921594 & 4.96511423174 & 0.0348857682557 \\ 6 & 397.383268336 & 5.98490122640 & 0.0150987735974 \\ 7 & 1089.61062510 & 6.99357568673 & 0.00642431327185 \\ 8 & 2972.94721365 & 7.99730906759 & 0.00269093240649 \\ 9 & 8094.07892206 & 8.99888807608 & 0.00111192392447 \\ 10 & 22016.4635234 & 9.99954579445 & 0.000454205553465 \\ 11 & 59863.1407045 & 10.9998162475 & 0.000183752470561 \\ 12 & 162742.790977 & 11.9999262640 & 0.0000737359850350 \\ 13 & 442400.391818 & 12.9999706149 & 0.0000293851457649 \\ 14 & 1202590.28408 & 13.9999883585 & 0.0000116415375921 \\ 15 & 3269002.37244 & 14.9999954114 & 0.00000458855586232 \\ 16 & 8886094.52049 & 15.9999981994 & 0.00000180056603754 \\ 17 & 24154935.7536 & 16.9999992962 & 0.000000703789907514 \\ 18 & 65659951.1373 & 17.9999997259 & 0.000000274139710557 \\ 19 & 178482281.963 & 18.9999998935 & 0.000000106453143645 \\ 20 & 485165175.410 & 19.9999999588 & 0.0000000412230741481 \\ 21 & 1318815713.48 & 20.9999999841 & 0.0000000159233771522 \\ 22 & 3584912824.13 & 21.9999999939 & 0.00000000613682984197 \\ 23 & 9744803423.25 & 22.9999999976 & 0.00000000236023232086 \\ 24 & 26489122105.8 & 23.9999999991 & 0.000000000906032291448 \end{array} } $$