 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2004/paper-2/8/a/i/solution

ln (1 + x) = x - x^{2} /2 + x^{3} /3 + O (x^{4})

and

(1 - x)^{- 1} = 1 + x + x^{2} + O (x^{3})

. Collecting the first three nonzero coefficients gives

\frac{ln ( 1 + x )}{1 - x} = x + \frac{x ^{2}}{2} + \frac{5 x ^{3}}{6} + O (x^{4}) .

The cubic coefficient is

1 - 1/2 + 1/3 = 5/6

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