 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2002/paper-2/10/i/solution

For every real

α

∣ cos (α n) / n^{2} ∣ \leq 1/ n^{2}

. The bounding P-series with exponent two converges. The comparison test for series therefore gives

absolute convergence for every α \in R .

Oscillations of cosine are unnecessary for convergence here.

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