More Numerical Integration Stability

Consider the following implicit numerical integration method. The $(n)$ superscript (with parentheses) in the sum denotes the $n$ th time derivative of the function $x$ , and the $n$ superscript (without parentheses) denotes raising $\Delta t$ to the power of $n$ .

$\large{x_k = x_{k-1} + \Sigma_{n=1}^{n=5} \, x_k^{(n)} (\Delta t)^n}$

Suppose we use this to discretely approximate the following continuous-time signal:

$\large{x(t) = e^{\alpha t}}$

In the above equation, $\alpha$ is a complex number. Define "stability" as the following being true for all processing steps:

$\large{\Big| \frac{x_k}{x_{k-1}} \Big| < 1}$

For $\Delta t = 1$ , how much area in the complex $\alpha$ plane is NOT included in the stability region?

Bonus: Plot the "divergence region" associated with the requested area