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1、数列极限2.1数列的极限一、数列的概念定义1设是一个以正整数集为定义域的函数,将其函数值按自变量n的大小顺序排成一列称为一个数列.数列中的每一个数叫做数列的项,第n项叫做数列的一般项或通项.数列也可表示为或2.一般说来,定义5中的N是随的变化而变化的,给定不同的,所确定的N一般也不同.3.定义5中“当时,有”的意思是从第N项的各项都满足4.数列极限的几何意义.就是对以A为中心,以任意小的正数为半径的邻域,总能找到一个N,从第项开始,以后的各项(无限多项)都落在邻域内,而在外,至多有N项(有限项).例1证明三
2、、数列极限的性质及收敛准则定理1(唯一性定理)若数列收敛,则其极限值必唯一.定理2(有界性定理)若数列收敛,则必是有界数列.若是无界数列,则发散,即不存在.定理3(保序性定理)设的极限存在,且则存在正整数N,当时,有推论1(保号性定理)设的极限存在,且(或),则存在正整数N,当时,有(或0).推论2设的极限存在,若(当时),则特别地,若(或),则(或).注:在推论2中即使是,也只能推出定理4(夹逼定理)设数列满足(当时),且,则例2求例3求即单调有界数列必有极限.数列必有极限;单调递减且有下界的数列必有极限
3、.定理5(单调有界数列收敛准则)单调递增且有上界的例4设,证明存在.例5设,证明存在.欣赏下列两个语段,并指出它们的技巧1.Briefasthepassageis,itstillarousesmyendlessimaginationtothepasttime.WereIinthepast,IwouldlikethatIstayedinBritaininthe19thcentury,whereandwhentheRevolutionofIndustrywasflourishing.Withtheinspiri
4、ngatmosphereofthecreativetimes,Iwouldtrymyutmosttodomyfavoritescientificexperiment,sinceIhavegreatinterestinscience.What’smore,thescientificloverswouldbeinvitedtojoinmylittleclubtoinventsomebrand-newinventionsandcreatenewdesignsofourdailylife’sproducts,coo
5、peratingtoputthetheorywelearnedfrombookstoourpracticallife.InthiswayIcouldgetinspirationofwisdomandresourcefulnesstobringconvenienceandimprovementstoourlife.2.ShouldIbegiventheopportunitiestotraveltothepast,IwouldliketogobacktoQingDynastyofearly20thcentury
6、whenChinawasinadisaster.WereIbackinthelateQingDynasty,IwouldmakeeveryendeavortorescueChinafromtheforeigninvadersandpreventitfrombeingbulliedwithoutresistance.Firstly,Iwouldteachmymoderntheoryandknowledgetothegovernors,instructingthemtofightagainsttheenemie
7、sandconstructpowerfularmies.Secondly,allthecitizenswouldbecalledontounitetogetherandresistothercountries’invasionbravely.What’smore,joiningsomeanti-invaderspartieswouldalsobemychoice.亮点句型写作指导除了虚拟语气外,有许多亮点句型可以应用到写作中,增分添彩。比喻AistoBwhatCistoD.Likethesunshine,k
8、indnesscanhelpthoseintrouble.Kindnessislikethelighthouse,whichcandirectthewayforthosewhoareindimsituation.排比IfIcouldgobacktothepastanddotheabovementionedthings,thelastthingmygrandparememberedwouldnotbetheindi