What are monotonic and bounded sequences? - StudyPug

We will learn that monotonic sequences are sequences which constantly increase or constantly decrease. We also learn that a sequence is bounded above if the ... Getthemostbyviewingthistopicinyourcurrentgrade.Pickyourcoursenow.LessonsOverview: MonotonicSequencesBoundedSequencesLessonsDifferencebetweenmonotonicandnon-monotonicsequences Showthatthefollowingsequences is monotonic.Isitanincreasingordecreasingsequence?{n2n^2n2}an=13na_n=\frac{1}{3^n}an​=3n1​{nn+1}n=1∞\{\frac{n}{n+1}\}_{n=1}^{\infty}{n+1n​}n=1∞​{1,1.5,2,2.5,3,3.5,...}Differencebetweenbounded,boundedabove,andboundedbelow Determinewhetherthesequencesareboundedbelow,boundedabove,both,orneitheran=n(−1)na_n=n(-1)^nan​=n(−1)nan=(−1)nn2a_n=\frac{(-1)^n}{n^2}an​=n2(−1)n​an=n3a_n=n^3an​=n3an=−n4a_n=-n^4an​=−n4Convegenceofsequences Arethefollowingsequencesconvergentaccordingto theorem7?{3n3}n=1∞\{\frac{3}{n^3}\}_{n=1}^{\infty}{n33​}n=1∞​{(−1)2n+12}n=1∞\{\frac{(-1)^{2n+1}}{2}\}_{n=1}^{\infty}{2(−1)2n+1​}n=1∞​{n}n=4∞\{\sqrt{n}\}_{n=4}^{\infty}{n​}n=4∞​Inthissection,wewillbetalkingaboutmonotonicandboundedsequences.Wewilllearnthatmonotonicsequencesaresequenceswhichconstantlyincreaseorconstantlydecrease.Wealsolearnthatasequenceisboundedaboveifthesequencehasamaximumvalue,andisboundedbelowifthesequencehasaminimumvalue.Ofcourse,sequencescanbebothboundedaboveandbelow.Lastly,wewilltakealookatapplyingtheorem7,whichwillhelpusdetermineifthesequenceisconvergent.Oneimportanttonotefromthetheoremisthateveniftheorem7doesnotapplytothesequence,thereisapossibilitythatthesequenceisconvergent.It'sjustthatthetheoremwillnotbeabletoshowit.Note Theorems: 1.Asequenceisincreasingifana_nan​an+1a_{n+1}an+1​foreveryn≥1n\geq1n≥1. 3.Ifasequenceisincreasingordecreasing,thenwecallitmonotonic. 4.AsequenceisboundedaboveifthereexistsanumberNsuchthatan≤Na_n\leqNan​≤Nforeveryn≥1n\geq1n≥1. 5.AsequenceisboundedbelowifthereexistsanumberMsuchthatan≥Ma_n\geqMan​≥Mforeveryn≥1n\geq1n≥1. 6.Asequenceisboundedifitisbothboundedaboveandboundedbelow. 7.Ifthesequenceisbothmonotonicandbounded,thenitisalwaysconvergent.Introductiontosequences2videosremainingtoday5practicequestionsremainingtodayBecomeamembertogetmore!JoinforFreeLearnMore