 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2/18/a/iii/solution

The standard Fourier series is

f (x) \sim \frac{a _{0}}{2} + \sum_{n = 1}^{\infty} (a_{n} cos n x + b_{n} sin n x),

a_{n} = \frac{1}{π} \int_{- π}^{π} f (x) cos n x d x (n \geq 0), b_{n} = \frac{1}{π} \int_{- π}^{π} f (x) sin n x d x (n \geq 1) .

These follow by taking inner products with the individual basis vectors and using their orthogonality and squared norms.

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