We explainwhatitmeansfortwosequencestobethesameandwha tismeantbythe thterm ofasequenceWealsoinvestigatethebehaviourofin64257nites equencesandseethattheymight tendtoplusorminusin64257nityortoareallimitorbehaveins omeotherway Inordertomasterthetechniqu ID: 26748
Download Pdf The PPT/PDF document "Limits of sequences mcTYsequences Inthis..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Limitsofsequences mc-TY-sequences-2009-1 Inthisunit,werecallwhatismeantbyasimplesequence,andintroducein nitesequences.Weexplainwhatitmeansfortwosequencestobethesame,andwhatismeantbythen-thtermofasequence.Wealsoinvestigatethebehaviourofin nitesequences,andseethattheymighttendtoplusorminusin nity,ortoareallimit,orbehaveinsomeotherway,Inordertomasterthetechniquesexplainedhereitisvitalthatyouundertakeplentyofpracticeexercisessothattheybecomesecondnature.Afterreadingthistext,and/orviewingthevideotutorialonthistopic,youshouldbeableto:usethenotationforasequenceintermsofaformulaforitsn-thterm;decidewhetherasequencetendstoin nity;decidewhetherasequencetendstominusin nity;decidewhetherasequencetendstoareallimit;decidewhetherasequencediverges;usethenotationforthelimitofasequence.Contents1.Introduction22.Somenotationforsequences33.Thebehaviourofin nitesequences3 www.mathcentre.ac.uk1c\rmathcentre2009 1.IntroductionAsimplesequenceisa nitelistofnumbers.Forexample,(1;3;5;:::;19)isasimplesequence.Sois(4;9;16;:::;81):Wecallthenumbersinthesequencethetermsofthesequence.Soinoursecondexamplewesaythatthe rsttermis4,thesecondtermis9,andsoon.Notallsequenceshavetobe nite.Wecanalsode nein nitesequences.Thesearelistsofnumbers,likesimplesequences,butthedi erenceisthatthetermsgoonforever.Wewritein nitesequenceslikethis:(2;5;8;:::):Ina nitesequence,thethreedotsinthemiddlefollowedbythe nalnumberjustmeanthatwehaveomittedsomeoftheterms.Butifyouseethreedotswithoutanythingafterthem,itmeansthatthesequencegoesonforever.Wesaytwosequencesarethesameifallthetermsarethesame.Thismeansthatthesequenceshavetocontainthesamenumbers,inthesameplaces,throughoutthesequence.So(1;2;3;4;:::)=(2;1;4;3;:::)eventhoughtheycontainthesamenumbers.Thepositionsofthenumbersaredi erent,sothesequencesarenotthesame.Our rsttwoexamplesofsequenceshaveobviousrulesforobtainingeachterm.Inthe rstexample,weobtainthen-thtermbytakingn,multiplyingby2,andtakingaway1.Wesaythatthen-thtermis2n 1.Similarly,then-thtermofthesecondsequenceis(n+1)2.Ourthirdexample,thein nitesequence,alsohasrulesforthen-thterm.Inthiscasewetaken,multiplyby3,andtakeaway1.Sothen-thtermis3n 1.Butnotallsequenceshaveclearrulesforthen-thterm.Forexample,thesequencethatstarts(p 3; 5;997;:::)isstillasequence,eventhoughitlooksrandom. KeyPointAsimplesequenceisa nitelistofnumbers,andanin nitesequenceisanin nitelistofnumbers.Thenumbersinthesequencearecalledthetermsofthesequence.Twosequencesarethesameonlyiftheycontainthesamenumbersinthesamepositions. www.mathcentre.ac.uk2c\rmathcentre2009 2.SomenotationforsequencesYoumighthavenoticedthatwehavewrittenoutoursequencesinbrackets.Forexample,wehavewritten(1;3;5;7;:::;19)forour rstsequence.Ifasequenceisgivenbyarulethenanother,moreconcise,waytodenotethesequenceistowritedowntheruleforthen-thterm,inbrackets.Sowemightwritethissequenceas(2n 1):Butwealsoneedtoshowhowmanytermsareincludedinthesequence,andwedothisbywritingtheindexofthe rsttermjustunderneaththebracket,andtheindexofthelasttermjustabove,likethis:(2n 1)10n=1:Similarly,wecouldwritethe nitesequence(4;9;16;25;:::;81)as((n+1)2)8n=1:Noticethatweusetwosetsofbracketsforthissequence.Theinnersetisusedtodenotetheformula(n+1)2forthen-thterm.Theoutersetindicatesthatwehaveasequence,includingallthetermswiththisformulafromn=1ton=8.Wecanusethesamenotationforin nitesequences.Ourexampleofanin nitesequencecanbewrittenas(3n 1)1n=1:Here,thesuperscript`1'isusedtoindicatethatthesequencegoesonforever. KeyPointWedenoteasequencebywritingthen-thterminbracketsandindicatinghowmanytermsareincludedinthesequence. 3.ThebehaviorofinnitesequencesWeshallnowconcentrateonin nitesequences.Itisoftenveryimportanttoexaminewhathappenstoasequenceasngetsverylarge.Therearethreetypesofbehaviourthatweshallwishtodescribeexplicitly.Thesearesequencesthat`tendtoin nity';sequencesthat`tendtominusin nity';sequencesthat`convergetoareallimit'. www.mathcentre.ac.uk3c\rmathcentre2009 Firstweshalllookatsequencesthattendtoin nity.Wesayasequencetendstoin nityif,howeverlargeanumberwechoose,thesequencebecomesgreaterthanthatnumber,andstaysgreater.Soifweplotagraphofasequencetendingtoin nity,thenthepointsofthesequencewilleventuallystayaboveanyhorizontallineonthegraph. n 510 any number we choose Herearesomeexamplesofsequencesthattendtoin nity.Atsomepointthissequencewillbegreaterthananynumberwechoose,andstaygreater,sothese-quencetendstoin nity. n 510 = 1 any number we choose Eventhoughthissequencesome-timesdecreases,itwillstilleven-tuallybecomegreater,andstaygreater,thananynumberwechoose. n 510 the sequence (1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6, ... ) any number we choose www.mathcentre.ac.uk4c\rmathcentre2009 Herearesomesequencesthatdonottendtoin nity.Forthissequenceweobtainthesecondtermbyadding100tothe rstterm.Then,togetthenextterm,weadd50,andthenweadd25,andsoon.Eachtimeweaddhalfasmuchaswedidbefore.Eventhoughthissequencestartsbyincreasingveryquickly,itwillneverbelargerthan200.Sowecanchooseanumberthatwillneverexceed. n 510 the sequence (0, 100, 150, 175, 187 200 Thissequencedoesgetlargerthananynumberwechoose.Butitdoesnotstaylarger,becauseitalwaysreturnstozero.Sowecannotsaythatthissequencetendstoin nity. n 510 the sequence (0, 1, 0, 2, 0, 3, 0, 4, 0, 5, ... ) any number we choose Thereissomenotationwecanuseifasequencetendstoin nity.Ifthesequence(xn)tendstoin nity,wewriteeitherxn!1asn!1orlimn!1(xn)=1: KeyPointAsequence(xn)tendstoin nityif,howeverlargeanumberwechoose,thesequencewilleventuallybecomegreaterthanthatnumber,andstaygreater. www.mathcentre.ac.uk5c\rmathcentre2009 Weshallnowlookatsequencesthattendtominusin nity.Wesayasequencetendstominusin nityif,howeverlargeanegativeanumberwechoose,thesequenceeventuallybecomeslessthanthatnumberandstayslessthanit.Soifweplotagraphofasequencetendingtominusin nity,thenthepointsofthesequencewilleventu-allystaybelowanyhorizontallineonthegraph. n 510 any number we choose Forexample,thissequencetendstominusin nity. n 510 any number we choose the sequence ( - n 3 ) n = 1 ¥ Butthissequencedoesnottendtominusin nity.Itdoesbecomelessthananynumberwechoose.Butitdoesnotstayless,becauseital-waysbecomespositiveagain.Sowecannotsaythatthissequencetendstominusin nity. n 510 any number we choose the sequence ( - 1, 1, - 2, 2, - 3, 3, ... ) www.mathcentre.ac.uk6c\rmathcentre2009 Thereissomenotationwecanuseifasequencetendstominusin nity.Ifthesequence(xn)tendstominusin nity,wewriteeitherxn! 1asn!1orlimn!1(xn)= 1: KeyPointAsequence(xn)tendstominusin nityif,howeverlargeanegativenumberwechoose,thesequencewilleventuallybecomelessthanthatnumber,andstayless. Finallyweshalllookatsequenceswithreallimits.Wesayasequencetendstoareallimitifthereisarealnumber,l,suchthatthesequencegetscloserandclosertoit.Wesaylisthelimitofthesequence.Thesequencegets`closerandclosertol'if,wheneverwedrawanintervalasnarrowaswechoosearoundl,thesequenceeventuallygetstrappedinsidetheinterval. n 510 l Noticethatitdoesnotmatterwhetherornotthesequenceeventuallytakesthevaluel.Wejustneedittogetascloseaswechoosetol.Thissequencegetsascloseasweliketozero.Sowesaythesequencetendstozeroasntendtoin nity. n 5 10 l 5 10 the sequence ( ) = 1 ¥ n 1 www.mathcentre.ac.uk7c\rmathcentre2009 Thissequencetendstothelimit3.Eventhoughthevaluesofthesequencesometimesmoveawayfrom3,theyeventuallystaywithinanyintervalaround3thatwechoose.Thenar-rowertheinterval,thefurtheralongthese-quencewemighthavetogobeforethevaluesbecometrappedwithintheinterval. n 510 3 Thereissomenotationwecanuseifasequencetendstoareallimit.Ifthesequence(xn)tendstothelimtl,wewriteeitherxn!lasn!1orlimn!1(xn)=l: KeyPointAsequence(xn)tendstoareallimitlif,howeversmallanintervalaroundlthatwechoose,thesequencewilleventuallytakevalueswithinthatinterval,andremainthere. Wesayasequenceisdivergentifitdoesnotconvergetoareallimit.Forexample,ifasequencetendstoin nityortominusin nitythenitisdivergent.Butnotalldivergentsequencestendtoplusorminusin nity.Forexample,thesequence(1;2;1;0; 1; 2; 1;0;1;2;1;0; 1; 2;:::)isdivergent,butitdoesnottendtoeitherin nityortominusin nity.Asequencelikethis,repeatingitselfoverandoveragain,iscalledaperiodicsequence. KeyPointAsequencethatdoesnotconvergetoareallimitiscalledadivergentsequence. www.mathcentre.ac.uk8c\rmathcentre2009 ExercisesDecidewhethereachofthefollowingsequencestendstoin nity,tendstominusin nity,tendstoareallimit,ordoesnottendtoalimitatall.Ifasequencetendstoareallimit,workoutwhatitis.1.(2n)1n=12.(1000 n)1n=13.n n+11n=14.(+p n)1n=15.sinn 41n=16.5 1 n1n=1Answers1.in nity2.minusin nity3.14.in nity5.nolimit6.5 www.mathcentre.ac.uk9c\rmathcentre2009