 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1/11/b/iii/solution

Use

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

e^{s i n x} = 1 + (x - \frac{x ^{3}}{6}) + \frac{x ^{2}}{2} + \frac{x ^{3}}{6} + O (x^{4}) .

The two cubic contributions cancel, leaving

e^{s i n x} = 1 + x + \frac{x ^{2}}{2} + O (x^{4}) .

Thus the coefficient of

x^{3}

is zero, even though the function itself is not an even function.

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