 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/ia/paper-1/9c/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

The absolute-value series can be grouped in pairs of nonnegative terms:

\sum_{n \geq 1} (∣ a_{2 n - 1} ∣ + ∣ a_{2 n} ∣) = \frac{5}{4} \sum_{n \geq 1} \frac{1}{n ^{2}} < \infty.

Indeed,

1/ n^{2} \leq \int_{n - 1}^{n} x^{- 2} d x

for

n \geq 2

, so these positive partial sums are bounded by two. Completeness of the real numbers proves their convergence. The triangle inequality bounds every late signed sum by the corresponding late absolute-value sum, so absolute convergence implies convergence. Thus the series converges absolutely.

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