 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2005/paper-1/12/b/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

Evenness removes every sine coefficient. In the period-two Fourier cosine series convention

f = a_{0} /2 + \sum_{n \geq 1} a_{n} cos (nπ x)

, the constant coefficient is

a_{0} = 2 \int_{0}^{1} (1 - x) d x = 1

a_{n} = 2 \int_{0}^{1} (1 - x) cos (nπ x) d x = \frac{2 [ 1 - ( - 1 ) ^{n} ]}{n ^{2} π ^{2}} .

f (x) = \frac{1}{2} + \frac{4}{π ^{2}} m = 0 \sum \infty \frac{cos (( 2 m + 1 ) π x )}{( 2 m + 1 ) ^{2}} .

The periodic function is continuous and piecewise smooth, so the Dirichlet-Jordan convergence theorem gives its value at every point. The summable absolute coefficient bound also gives uniform convergence by the Weierstrass M-test.

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