 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/i/solution

The positive polynomial root is

x_{*} = ε + 1 + ε^{2}

. Its Taylor expansion near zero is

x_{*} = 1 + ε + \frac{1}{2} ε^{2} - \frac{1}{8} ε^{4} + O (ε^{6}) .

The starting error is therefore

x_{0} - x_{*} = - ε - \frac{1}{2} ε^{2} + O (ε^{4})

. Consequently

∣ x_{0} - x_{*} ∣ \sim ∣ ε ∣, n_{0} = 1.

For signed

ε

, the magnitude scaling should be written with

∣ ε ∣

; this is the precise interpretation of the proportionality in the question.

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