 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1/15/c/i/solution

sin x = x - x^{3} /6 + O (x^{5})

and

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

. Multiplying through the cubic order gives

\frac{sin x}{( 1 + x ) ^{2}} = x - 2 x^{2} + \frac{17}{6} x^{3} + O (x^{4}) .

These are the first three nonzero terms; the constant term vanishes.

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