 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/i/solution

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

Put

h = x - 1

. The Taylor series of the natural logarithm gives the first four nonzero terms:

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

The corresponding infinite series converges for

∣ h ∣ < 1

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