 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2016/discrete-mathematics-continuous-mathematics-and-probability-resit/6/f/solution

For

1 < x_{0} < 2

x_{1} - 2 = \frac{( x _{0} - 2 ) ^{2}}{2 ( x _{0} - 1 )} > 0.

Thus the first step overshoots the root to $x_{1} > 2$ . Part (e) then applies from index

1

: the remaining sequence decreases to

2

. The first step can be very large when

x_{0}

is close to

1

, where the derivative vanishes, but every starting value strictly between

1

and

2

still gives convergence to $2$ . The entire sequence is not decreasing, since

x_{1} > 2 > x_{0}

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