数学专业英语7.docx

数学专业英语7.docx

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数学专业英语7

MathematicalEnglish

Dr.XiaominZhang

Email:

zhangxiaomin@

§2.7SequencesandTheirLimits

TEXTAThedefinitionofsequences

IneverydayusageoftheEnglishlanguage,thewords“sequence”and“series”aresynonyms,andtheyareusedtosuggestasuccessionofthingsoreventsarrangedinsomeorder.Inmathematicsthesewordshavespecialtechnicalmeanings.Theword“sequence”isemployedasinthecommonuseofthetermtoconveytheideaofasetofthingsarrangedinorder,buttheword“series”isusedinasomewhatdifferentsense.Theconceptofasequencewillbediscussedinthissection,andserieswillbedefinedinSection11.

Ifforeverypositiveintegernthereisassociatedarealorcomplexnumberan,thentheorderedset

a1,a2,a3,…,an,…

issaidtodefineaninfinitesequence.Theimportantthinghereisthateachmemberofthesethasbeenlabeledwithanintegersothatwemayspeakofthefirstterma1,thesecondterma2,and,ingeneral,thenthterman.Eachtermanhasasuccessoran+1andhencethereisno”last”term.

Themostcommonexamplesofsequencescanbeconstructedifwegivesomeruleorformulafordescribingthenthterm.Thus,forexample,theformulaan=1/ndefinesasequencewhosefirstfivetermsare

1,1/2,1/3,1/4,1/5.

Sometimestwoormoreformulasmaybeemployedas,forexample,

a2n-1=1,a2n=2n2,

thefirstfewtermsinthiscasebeing

1,2,1,8,1,18,1,32,1.

Anothercommonwaytodefineasequenceisbyasetofinstructionswhichexplainshowtocarryonafteragivenstart.Thuswemayhave

a1=a2=1,an+1=an+an-1forn2.

ThisparticularruleisknownasarecursionformulaanditdefinesafamoussequencewhosetermsarecalledtheFibonaccinumbers.Thefirstfewtermsare

1,1,2,3,5,8,13,21,34

Inanysequencetheessentialthingisthattherebesomefunctionfdefinedonthepositiveintegerssuchthatf（n）isthenthtermofthesequenceforeachn=1,2,3,…..Infact,thisisprobablythemostconvenientwaytostateatechnicaldefinitionofsequence.

DEFINITIONAfunctionfwhosedomainisthesetofallpositiveintegers1,2,3,…iscalledaninfinitesequence.Thefunctionvaluef（n）iscalledthenthtermofthesequence.

Therangeofthefunction（thatis,thesetoffunctionvalues）isusuallydisplayedbywritingthetermsinorder,thus:

f

（1）,f

（2）,f（3）,…,f（n）,…

Forbrevity,thenotation{f（n）}isusedtodenotethesequencewhosenthtermisf（n）.Veryoftenthedependenceonnisdenotedbyusingsubscripts,andwewritean,sn,xn,un,orsomethingsimilarinsteadoff（n）.Unlessotherwisespecified,allsequences（suchasarithmeticprogressionandgeometricprogression）inthischapterareassumedtohaverealorcomplexterms.

Notations

SeriesAseriesisaninfiniteorderedsetoftermscombinedtogetherbytheadditionoperator.Theterminfiniteseriesissometimesusedtoemphasizethefactthatseriescontainaninfinitenumberofterms.Theorderofthetermsinaseriescanmatter,sincetheRiemannseriestheoremstatesthat,byasuitablerearrangementofterms,aso-calledconditionallyconvergentseriesmaybemadetoconvergetoanydesiredvalue,ortodiverge.

Ifthedifferencebetweensuccessivetermsofaseriesisaconstant,thentheseriesissaidtobeanarithmeticseries.Aseriesforwhichtheratioofeachtwoconsecutivetermsak+1/akisaconstantfunctionofthesummationindexkiscalledageometricseries.

arithmeticprogressionAnarithmeticprogression,alsoknownasanarithmeticsequence,isasequence{a0+nd},n=0,1,2,…,suchthatthedifferencesbetweensuccessivetermsisaconstantd,wherea0anddarecalledinitialtermandcommondifferencerespectively.

geometricprogressionAlsoknownasageometricsequence,isasequence{an},n=0,1,2,…,suchthateachtermisgivenbyamultiplerofthepreviousone.Ifthemultiplierisr,thenthekthtermisgivenby

an=ran-1=r2an-2=a0rn

Takinga0=1givesthesimplespecialcasean=rn.

TEXTBThelimitofasequence

Themainquestionweareconcernedwithhereistodecidewhetherornotthetermsf（n）tendtoafinitelimitasnincreasesinfinitely.Totreatthisproblem,wemustextendthelimitconcepttosequence.Thisisdoneasfollows.

DEFINITIONAsequence{f（n）}issaidtohavelimitLif,foreverypositivenumber,thereisanotherpositivenumberN（whichmaydependon）suchthat

|f（n）-L|

Inthiscase,wesaythesequence{f（n）}convergestoLandwewrite

limnf（n）=L,orf（n）Lasn.

Asequencewhichdoesnotconvergeiscalleddivergent.

Inthisdefinitionthefunctionvaluesf（n）andthelimitLmayberealorcomplexnumbers.IffandLarecomplex,wemaydecomposethemintotheirrealandimaginaryparts,sayf=u+ivandL=a+ib.Thenwehavef（n）-L=u（n）-a+i[v（n）-b].Theinequalities

|u（n）-a||f（n）-L|and|v（n）-b||f（n）-L|

showthattherelationf（n）Limpliesu（n）aandv（n）basn.Conversely,theinequality

|f（n）-L||u（n）-a|+|v（n）-b|

showsthatthetworelationsandu（n）aandv（n）bimpliesf（n）Lasn.Inotherwords,acomplex–valuedsequencefconvergesifandonlyifboththerealpartuandtheimaginarypartvconvergeseparately,inwhichcasewehave

limnf（n）=limnu（n）+ilimnv（n）.

Itisclearthatanyfunctiondefinedforallpositiverealxmaybeusedtoconstructasequencebyrestrictingxtotakeonlyintegervalues.ThisexplainsthestronganalogybetweenthedefinitionjustgivenandtheoneinSection6.4formoregeneralfunctions.Theanalogycarriesovertoinfinitelimitsaswell,andweleaveitforthereadertodefinethesymbols

limnf（n）=+andlimnf（n）=-

aswasdoneinSection6.5whenfisreal-valued.Iffiscomplex,wewritef（n）asnif|f（n）|+.

Thephrase“convergentsequence”isusedonlyforasequencewhoselimitisfinite.Asequencewithaninfinitelimitissaidtodiverge.Thereare,ofcourse,divergentsequencesthatdonothaveinfinitelimits.Examplesaredefinedbythefollowingformulas:

f（n）=（-1）n,f（n）=sin（n/2）,f（n）=（-1）n（1+1/n）,f（n）=ei（n/2）.

Thebasicrulesfordealingwithlimitsofsums,products,etc.,alsoholdforlimitsofconvergentsequences.Thereadershouldhavenodifficultyinformulatingthesetheoremsforhimself.TheirproofsaresomewhatsimilartothosegiveninSection3.5.

SUPPLEMENTFibonacciandFibonaccinumbers

Fibonacci,LeonardodaPisa（1170-1240）ItalianmathematicianwhowasthefirstgreatWesternmathematicianafterthedeclineofGreekscience.Thesonofamerchant,FibonaccidrewthemotivationtomathematicalinquiryfromhiscommercialtripstotheOrient.ItwassomewherebetweenBarbary（Maghreb）andConstantinople（nowIstanbul）thathegotacquaintedwiththeHindu-ArabicnumbersystemanddiscovereditsenormouspracticaladvantagescomparedtotheRomannumerals,whichwerestillcurrentinWesternEurope.

Performingeventhesimplestarithmeticaloperationswithanon-positionalnotationwasadifficultendeavor:

forthistaskthemerchantswereforcedtoresorttotheabacus,adevicewherethenumberswererepresentedbymovingballs.FibonacciexposedthenewalternatecomputingmethodbasedonwrittenalgorithmsratherthanoncountingobjectsinhisLiberAbaci,firstissuedin1202.

Thebookbeganwithapresentationofwhathecalledtheten"Indianfigures"（0,1,2,...,9）.Itwasintendedasanalgebramanualforcommercialuse,andexplainedthearithmeticalrulesusingnumericalexamplesderived,forexample,frommeasureandcurrencyconversion,whichweretranslatedintoproportionsandsolvedbymultiplication（ruleofthree）.Theso-calledFibonaccisequencearoseinthisbookfromaconcretequestionconcerningthegrowthofarabbitpopulation.Geometricprogressionsalsoappearedinproblemsrelatedtolegacyandinterest.

ThetreatisePracticaGeometriae,publishedin1225,ismainlyinspiredbyGreekmathematics;itcontainstheoremsfromEuclid'sElementsandalsoHeron'sformulafortheareaofatriangle.

FibonaccidistinguishedhimselfinthemathematicalcompetitionsproposedatthecourtofEmperorFrederickIIofHohenstaufen,KingoftheTwoSicilies,whohadhisroyalseatinPalermo.HisstrikingabilityinsolvingalgebraicequationsofhigherdegreeclearlyemergesfromhisworksentitledLiberQuadratorumandFlos,bothofwhichappearedin1225.Thefirstcontainsformulasandequationsinvolvingperfectsquares,thesecondowesitsfametotheirrationalsolutionofacubicequation,whichFibonaccideterminedwithanaccuracyof10-9.Mostofhissolvingtechniquesseemtobebasedonthealgebraicworksofal-Khwarizmi.

Al-Khwarizmi,MuhammedibnMusa（ca.780-ca.850）PersianmathematicianwhowroteatreatiseaboutalgebraicmethodswhoseArabictitlewasdistortedintothewordalgebra.HeusedtheHindunumberzero,whichwasthenintroducedtoEuropewhenAl-Khwarizmi'sworkwastranslatedbyFibonacci.

Fibonacciinitiatedthetraditionofthemaestrid'abaco,expertsinpracticalalgebraandarithmetic,whoflourishedinItalyduringthe14thcentury,andcanbeconsideredastheforerunnersofCardano,Tartaglia,andFerrari.

FibonaccisequenceTheFibonaccinumbersofthesequenceofnumbersFn,canbeviewedasaparticularcaseoftheFibonaccipolynomialsFn（x）withFn=Fn

（1）.Theysatisfytherecurrencerelation,

Fn=Fn-2+Fn-1

forn=3,4,...,withF1=F2=1.ThefirstfewFibonaccinumbersforn=1,2,...are1,1,2,3,5,8,13,21,34,55,89,144,233,377,....

TheFibonaccinumbersgivethenumberofpairsofrabbitsnmonthsafterasinglepairbeginsbreeding（andnewlybornbunniesareassumedtobeginbreedingwhentheyaretwomonthsold）,asfirstdescribedbyFibonacciinhisbookLiberAbaci.

TheratiosofsuccessiveFibonaccinumbers