Paper 1, Section II, F

Analysis I | Part IA, 2011

(a) State, without proof, the ratio test for the series n1an\sum_{n \geqslant 1} a_{n}, where an>0a_{n}>0. Give examples, without proof, to show that, when an+1<ana_{n+1}<a_{n} and an+1/an1a_{n+1} / a_{n} \rightarrow 1, the series may converge or diverge.

(b) Prove that k=1n11klogn\sum_{k=1}^{n-1} \frac{1}{k} \geqslant \log n.

(c) Now suppose that an>0a_{n}>0 and that, for nn large enough, an+1an1cn\frac{a_{n+1}}{a_{n}} \leqslant 1-\frac{c}{n} where c>1c>1. Prove that the series n1an\sum_{n \geqslant 1} a_{n} converges.

[You may find it helpful to prove the inequality log(1x)<x\log (1-x)<-x for 0<x<10<x<1.]

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