 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2006/paper-1/9/b/ii/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

For the arctangent, put

h = x - 1

(d / d h) arctan (1 + h) = 1/ (2 + 2 h + h^{2}) = \frac{1}{2} - \frac{h}{2} + \frac{h ^{2}}{4} + O (h^{4})

. Integrating and using

arctan 1 = π /4

gives

arctan x = \frac{π}{4} + \frac{h}{2} - \frac{h ^{2}}{4} + \frac{h ^{3}}{12} + O (h^{5}) .

The constant counts as the first nonzero term. The quartic coefficient vanishes; the next nonzero term is

- h^{5} /40

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