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 Incoming links: Squares of a summable positive sequence

Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 3F / a / Solution Created 2026-09-24 Updated 2026-09-25

Convergence of

\sum a_{n}

implies

a_{n} \to 0

. Hence

0 < a_{n} \leq 1

for all sufficiently large

n

, and then

a_{n}^{2} \leq a_{n}

. The comparison test, with the finitely many initial terms treated separately, proves that

n = 1 \sum \infty a_{n}^{2} < \infty .

This is the squares of a summable positive sequence lemma.

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