 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2001/paper-1/9/d/solution

Use the real logarithm near zero and subtract its two Taylor series:

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

Thus the first three nonzero terms are

- x - \frac{5}{2} x^{2} - \frac{1}{3} x^{3} .

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