Final answer:
The iteration method Xk+1 = xk + e^-4 doesn't guarantee convergence by the fixed point theorem because g'(x) = 1, not less than one. The method could still potentially converge monotonically, and if convergent, it would likely show linear convergence due to the constant value added at each step. To determine the exact behavior and rate of convergence, further analysis specific to the function and root would be needed.
Step-by-step explanation:
To prove convergence for the iteration method Xk+1 = xk + e^-4, we can examine whether the absolute value of the derivative of the function at the fixed point is less than one. The fixed point of the given iteration method is a solution to (x-1)e^x - 1 = 0. If we assume x* is this solution, then the function defining the iteration method is g(x) = x + e^-4. Since g'(x) = 1, which is not less than one, this particular iteration method does not guarantee convergence by the usual fixed point theorem.
However, because e^-4 is a constant and g'(x) = 1 does not depend on x, each iteration step adds a fixed small constant to the approximation. This means that the sequence {xk} will be monotonically increasing or decreasing, allowing us to examine specific bounds to determine if it converges to the root of the equation. Nevertheless, since convergence cannot be guaranteed by the fixed point theorem, let's consider the average rate of convergence if we assume the method converges to a root x*. The average rate of convergence can be found by taking the limit as k approaches infinity of the absolute value of (x_(k+1) - x*) / (xk - x*)^p, where p is the order of convergence, and trying to solve for p. Since the method adds a constant value at each step, it suggests a linear convergence, so we may assume p = 1.
If this were a convergent iteration function with a linear rate of convergence, the value of e^-4 would dictate the rate. However, because g'(x) = 1, as mentioned earlier, this particular iteration method does not have a guaranteed rate of convergence. Further analysis of this iteration sequence in the context of the specific function it is trying to find a root of would be required to determine its behavior more precisely.