 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-v/4/b/solution

f

is written

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

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

By part (a)(i), any other interval of length

2 π

2 π

. The notation

\sim

associates

f

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