 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/ia/paper-1/11f/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

s \leq 0

, the terms

n^{- s}

do not tend to zero, so the series diverges by the term test for divergence. If

s > 0

, the terms form a nonnegative decreasing sequence, and the Cauchy condensation test gives the condensed series

\sum_{k = 0}^{\infty} 2^{k} (2^{k})^{- s} = \sum_{k = 0}^{\infty} 2^{k (1 - s)} .

This geometric series converges exactly when

1 - s < 0

. Hence the p-series criterion is

n = 1 \sum \infty n^{- s} converges ⟺ s > 1 .

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