 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-1/19/b/ii/solution

x_{k + 1} = x_{k} - \frac{x _{k}^{2} - 2 ε x _{k} - 1}{2 ( x _{k} - ε )} = \frac{x _{k}^{2} + 1}{2 ( x _{k} - ε )} .

x_{0} = 1

, it gives

x_{1} = 1/ (1 - ε)

. Its geometric series expansion is

1 + ε + ε^{2} + ε^{3} + O (ε^{4})

. Subtracting the expansion of the polynomial root yields

x_{1} - x_{*} = \frac{1}{2} ε^{2} + ε^{3} + O (ε^{4}), ∣ x_{1} - x_{*} ∣ \sim \frac{1}{2} ∣ ε ∣^{2}, n_{1} = 2.

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