Final answer:
The fixed point iteration given by Xk+1 = sin(xk+π/2)/2 will converge due to the bounded derivative of the function g(x) between -0.5 and 0.5. The iterations converge to a fixed point where 2x equals sin(x+π/2), as long as the initial value is chosen appropriately.
Step-by-step explanation:
When analyzing whether a fixed point iteration converges, and to which value it converges, one must look at the function g(x) and its properties. For the iteration Xk+1 = g(Xk) where g(x) = sin(x+π/2)/2, the fixed point theorem can be applied if g(x) is continuous and differentiable, and its derivative is bounded by a value strictly less than 1 in magnitude on an interval containing the fixed point.
In this case, since sin(x) oscillates between -1 and 1, g(x) will oscillate between -0.5 and 0.5. Thus, |g'(x)| = |cos(x+π/2)/2| which oscillates between 0 and 0.5. This means the absolute value of the derivative of g(x) is always less than 1, satisfying the conditions for convergence.
The fixed point will be the value x such that x = sin(x+π/2)/2, or equivalently 2x = sin(x+π/2). The iterations should converge to this fixed point as long as the initial guess is within the basin of attraction to the fixed point. Without further analytical or numerical methods to find this precise value, we can assume that a fixed point exists and the iterations will converge to it based on the properties of g(x).