 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2/8/solution

sin 3 x = 3 x - \frac{9}{2} x^{3} + O (x^{5})

and

cos x = 1 - \frac{1}{2} x^{2} + O (x^{4})

f (x) = 3 x - 6 x^{3} + O (x^{5}) .

The same terms follow from the two sine harmonics in the Fourier series found above.

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