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Question Number 165081 by mathocean1 last updated on 25/Jan/22
Givena;b>0and∀n∈N,un;vn>0. {u0=aun+1=unvnand{v0=bvn+1=12(un+vn). Showthatthesequencesunandun areconvergentandhavethesame limitl(liscalledthearithmetico−geometriclimit).
Answered by aleks041103 last updated on 27/Jan/22
weknow ab⩽12(a+b) ⇒0<u1⩽v1 also ukvk=uk+1⩽vk+1=12(vk+uk) supposeuk⩽vk(trueifvk,uk⩾0) ⇒uk+1=vkuk⩾ukuk=uk ⇒uk+1⩾uk ⇒uk⩾u1>0⇒vk,uk>0 ⇒uk⩽vkistrue ⇒uk+1=vkuk⩽vkvk=vk ⇒uk+1⩽vk Now vk+1=12(vk+uk)⩾12(uk+uk)=uk ⇒vk+1⩾uk vk+1=12(vk+uk)⩽12(vk+vk)=vk ⇒vk+1⩾vk Fromallofthiswehave: u1⩽uk⩽uk+1⩽vk⩽v1 ⇒u1⩽uk⩽v1⇒{un}isbounded. alsouk+1⩾uk⇒{un}isincreasing ⇒{un}converges. u1⩽uk⩽vk+1⩽vk⩽v1 ⇒u1⩽uk⩽v1⇒{vn}isbounded. alsovk+1⩽vk⇒{vn}isdecreasing ⇒{vn}converges. uk+1=ukvk Double subscripts: use braces to clarifyDouble subscripts: use braces to clarify since{uk}converges,limk→∞uk+1uk=1 Double subscripts: use braces to clarifyDouble subscripts: use braces to clarify
Commented bymathocean1 last updated on 27/Jan/22
thankyouverymuch.
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