 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/c/solution

Evaluate at

x = π /2

, where the original expression gives

f (π /2) = 1 - π^{2} /4

1 - \frac{π ^{2}}{4} = 1 - \frac{π ^{2}}{6} + \sum_{m = 1}^{\infty} \frac{( - 1 ) ^{m}}{m ^{2}} .

Rearranging proves

m = 1 \sum \infty \frac{( - 1 ) ^{m}}{m ^{2}} = - \frac{π ^{2}}{12} .

The convergence at this point is justified by the continuity properties noted in part (b).

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