 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/15/a/i/solution

Expand the natural logarithm and reciprocal separately:

ln (2 + x) = ln 2 + \frac{x}{2} - \frac{x ^{2}}{8} + O (x^{3}), \frac{1}{2 - x} = \frac{1}{2} (1 + \frac{x}{2} + \frac{x ^{2}}{4} + O (x^{3})) .

Multiplication gives the first three nonzero terms of the Maclaurin series:

\frac{ln ( 2 + x )}{2 - x} = \frac{ln 2}{2} + \frac{1 + ln 2}{4} x + \frac{1 + 2 ln 2}{16} x^{2} + O (x^{3}) .

The local expansions converge for

∣ x ∣ < 2

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