Final answer:
The limit of the given sequence is 0. The order of convergence is 1, and the rate of convergence can be found by calculating the value of a limit.
Step-by-step explanation:
To find the order of convergence and rate of convergence for the given sequence x(k+1)=1−cos(x(k)), we need to first compute the limit of the sequence. Let's calculate the first few terms:
x(0) = 0.5
x(1) = 1 − cos(0.5) ≈ 0.8776
x(2) = 1 − cos(0.8776) ≈ 0.6547
x(3) = 1 − cos(0.6547) ≈ 0.7915
As we continue the calculations, we can observe that the terms of the sequence are getting close to 0. Let's calculate a few more terms:
x(4) ≈ 0.7067
x(5) ≈ 0.7653
x(6) ≈ 0.7172
x(7) ≈ 0.7470
From these calculations, it is clear that the terms of the sequence are converging to 0. Therefore, the limit of the sequence is 0. Now, let's determine the order of convergence:
If the sequence converges to the limit L with order p, then
lim (k→∞) [x(k+1) - L] / [x(k) - L]^p = C
where C is a non-zero constant. By observing the behavior of the terms as k approaches infinity, we can conclude that the order of convergence for this sequence is 1. Since p = 1, we can find the rate of convergence:
lim (k→∞) [x(k+1) - L] / [x(k) - L] = C
By calculating the value of the limit, we can determine the rate of convergence.