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Question Number 87308 by Rio Michael last updated on 03/Apr/20

A sequence (U_n ) is defined reculsively as U_o = (1/2) and U_(n+1) = (2/(1 + U_n )) for n ∈ N a) Show by mathematical induction that all terms in the sequence are positive. b) Given that the sequence (U_n ) is convergent, show that the limit,l, is a solution to the equation x^2 + x−2 = 0. Hence find l c) Given that (V_n ) is a sequence of general term such that V_n = ((U_n −1)/(U_n +2)) , ∀ n ∈ N. show that (V_n ) is convergent and determine its limit. hence deduce the convergence of the sequence (U_n ). Please recommend me textbooks for this topic even youtube vids please

$$\:\mathrm{A}\:\mathrm{sequence}\:\left({U}_{{n}} \right)\:\mathrm{is}\:\mathrm{defined}\:\mathrm{reculsively}\:\mathrm{as}\: \\ $$$$\:{U}_{{o}} \:=\:\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{and}\:{U}_{{n}+\mathrm{1}} \:=\:\frac{\mathrm{2}}{\mathrm{1}\:+\:{U}_{{n}} }\:\mathrm{for}\:\mathrm{n}\:\in\:\mathbb{N} \\ $$$$\left.\:\mathrm{a}\right)\:\mathrm{Show}\:\mathrm{by}\:\mathrm{mathematical}\:\mathrm{induction}\:\mathrm{that}\:\mathrm{all}\:\mathrm{terms}\:\mathrm{in}\:\mathrm{the}\:\mathrm{sequence} \\ $$$$\:\:\:\:\:\mathrm{are}\:\mathrm{positive}. \\ $$$$\left.\mathrm{b}\right)\:\mathrm{Given}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sequence}\:\left({U}_{{n}} \right)\:\mathrm{is}\:\mathrm{convergent},\:\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{limit},{l},\:\mathrm{is} \\ $$$$\:\:\:\:\mathrm{a}\:\mathrm{solution}\:\mathrm{to}\:\mathrm{the}\:\mathrm{equation}\:{x}^{\mathrm{2}} \:+\:{x}−\mathrm{2}\:=\:\mathrm{0}.\:\mathrm{Hence}\:\mathrm{find}\:{l} \\ $$$$\left.\:\mathrm{c}\right)\:\:\mathrm{Given}\:\mathrm{that}\:\left({V}_{{n}} \right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{sequence}\:\mathrm{of}\:\mathrm{general}\:\mathrm{term}\:\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\:\:\:\:\:{V}_{{n}} \:=\:\frac{{U}_{{n}} −\mathrm{1}}{{U}_{{n}} +\mathrm{2}}\:,\:\forall\:{n}\:\in\:\mathbb{N}. \\ $$$$\:\:\mathrm{show}\:\mathrm{that}\:\left({V}_{{n}} \right)\:\mathrm{is}\:\mathrm{convergent}\:\mathrm{and}\:\mathrm{determine}\:\mathrm{its}\:\:\mathrm{limit}. \\ $$$$\mathrm{hence}\:\mathrm{deduce}\:\mathrm{the}\:\mathrm{convergence}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sequence}\:\left({U}_{{n}} \right). \\ $$$$\:\:{Please}\:{recommend}\:{me}\:{textbooks}\:{for}\:{this}\:{topic}\:{even}\:{youtube}\:{vids} \\ $$$${please} \\ $$$$ \\ $$

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