 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2006/paper-2/10/e/solution

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

Put

p = k - 1 > 0

. The preceding inequality bounds

n^{- k} < [(n - 1)^{- p} - n^{- p}] / p

for

n \geq 2

. For

N \geq 2

, summing the telescoping series gives

\sum_{n = 1}^{N} n^{- k} < 1 + \frac{1 - N ^{- (k - 1)}}{k - 1} < 1 + \frac{1}{k - 1} .

These nonnegative partial sums increase and are bounded, so the monotone convergence theorem for sequences proves convergence for every $k > 1$ . This is the telescoping bound for a p-series, an alternative to the integral test.

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