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 Incoming links: Ratio comparison test

Past exam of the mathematics course of the University of Cambridge / 2004 / ia / Paper 1 / 12F / i / Solution Created 2026-09-24 Updated 2026-10-14

Rearranging the assumed positive-term ratio inequality gives

\frac{a _{n + 1}}{b _{n + 1}} \leq \frac{a _{n}}{b _{n}} .

Induction therefore yields

a_{n} / b_{n} \leq a_{1} / b_{1}

for every

n \geq 1

. In particular

0 \leq \sum_{n = 1}^{N} a_{n} \leq \frac{a _{1}}{b _{1}} \sum_{n = 1}^{N} b_{n} \leq \frac{a _{1}}{b _{1}} \sum_{n = 1}^{\infty} b_{n} .

The positive partial sums of the first series are nondecreasing and bounded above, so the bounded monotone sequence theorem proves that they converge. This is the ratio comparison test, and establishes convergence of $\sum a_{n}$ .

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