MTH 253 Calculus (Other Topics)Does the sequence converge?TerminologyIs (strictly) monotone?Slide 5Properties that Hold EventuallyConvergence of Monotone SequencesSlide 8Slide 9MTH 253Calculus (Other Topics)Chapter 10 – Infinite SeriesSection 10.2 – Monotone SequencesCopyright © 2006 by Ron Wallace, all rights reserved.Does the sequence converge?What a sequence converges to is not always important, but does it converge to something?This Section:Determining if a sequence converges without finding the actual limit.Terminology{ }1nna�=1Strictly Increasing: , for all n na a n+<1Strictly Decreasing: , for all n na a n+>1Increasing: , for all n na a n+�1Decreasing: , for all n na a n+�Monotone Increasing OR Decreasing�Strictly Monotone Strictly Increasing OR Decreasing�How would you describe ?{ }15n�=Is (strictly) monotone?{ }1nna�=Two TestsDifference Between Terms11110 strictly inc.0 strictly dec.0 increasing0 decreasingn nn nn nn na aa aa aa a++++- > �- < �- � �- � �Ratio of Successive Terms11111 strictly inc.1 strictly dec.1 increasing1 decreasingn nn nn nn na aa aa aa a++++> �< �� �� �11Determine the Behavior of 1nn�=� �-� ��(use both methods)Is (strictly) monotone?{ }1nna�=A Third Test using Derivatives'( ) 0 strictly inc.'( ) 0 strictly dec.'( ) 0 increasing'( ) 0 decreasingf xf xf xf x> �< �� �� �11Determine the Behavior of 1nn�=� �-� ��Let ( ) be any functionwhere 1 and ( )whenever .nf xx f x ax n� ==Properties that Hold EventuallyIf a finite number of the terms from the beginning of a sequence are discarded and the resulting sequence has a property, then the original sequence has that property eventually.Example2 3 4 35, 2, 9,3, , , ,..., ,...3 4 5 2nn---What can be said about:The sequence is eventually strictly increasing.(discard the first 4 terms)Convergence of Monotone SequencesIf a sequence is eventually increasing, then there are two possibilities:.1 M (upper bound) where all an M and the sequence converges to a value L M.2. No upper bound and the sequence approaches infinity as n.Note: In the first case, finding M guarantees convergence without the need to find L.Convergence of Monotone SequencesIf a sequence is eventually decreasing, then there are two possibilities:1. M (lower bound) where all an M and the sequence converges to a value L M.2. No lower bound and the sequence approaches negative infinity as n.Note: In the first case, finding M guarantees convergence without the need to find L.Convergence of Monotone SequencesExamples2!nn� �� ��Eventually decreasing and always positive. (why?)Therefore it converges. (why?)21nn� �-� ��Eventually increasing but no upper bound. (why?)Therefore it diverges.
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