Paper 1, Section II, E

Analysis I | Part IA, 2019

Let (an)\left(a_{n}\right) and (bn)\left(b_{n}\right) be sequences of positive real numbers. Let sn=i=1nais_{n}=\sum_{i=1}^{n} a_{i}.

(a) Show that if an\sum a_{n} and bn\sum b_{n} converge then so does (an2+bn2)1/2\sum\left(a_{n}^{2}+b_{n}^{2}\right)^{1 / 2}.

(b) Show that if an\sum a_{n} converges then anan+1\sum \sqrt{a_{n} a_{n+1}} converges. Is the converse true?

(c) Show that if an\sum a_{n} diverges then ansn\sum \frac{a_{n}}{s_{n}} diverges. Is the converse true?

[[ For part (c), it may help to show that for any NNN \in \mathbb{N} there exist mnNm \geqslant n \geqslant N with

an+1sn+1+an+2sn+2++amsm12.]\left.\frac{a_{n+1}}{s_{n+1}}+\frac{a_{n+2}}{s_{n+2}}+\ldots+\frac{a_{m}}{s_{m}} \geqslant \frac{1}{2} .\right]

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