本科毕业论文-沃利斯公式的证明及其应用

盐城师范学院毕业论文沃利斯公式的证明及其应用学生姓名学院数学科学学院专业数学与应用数学班级10（2）班学号10211255指导教师韩诚2014年5月25日沃利斯公式的证明及其应用摘要WALLIS公式在求EULERPOISSON积分和推导STIRLING公式的过程中扮演了很重要的角色近几年来，国内很多数学分析的教材都引入WALLIS公式，但教材中关于其应用的论述很少本文针对WALLIS公式的证明并将WALLIS公式进行两个简单推广，从数列极限计算、积分计算以及级数收敛性判断几个方面探讨WALLIS公式的应用，为微积分教学提供有意义的素材和思路【关键词】WALLIS公式；极限；积分PROOFANDITSAPPLICATIONSOFWALLISFORMULAABSTRACTTHEFORMULAOFWALLISPLAYSANIMPORTANTROLEINTHEPROCESSTOOBTAINTHEEULERPOISSONINTEGRALANDTHEDERIVATIONOFSTIRLINGFORMULAINRECENTYEARS,MANYDOMESTICANALYSISMATHEMATICSTEXTBOOKSINTOWALLISFORMULA,BUTLITTLEABOUTTHEAPPLICATIONSOFTHETEACHINGMATERIALTHISPAPERPROVESTHATTHELITTLEWALLISFORMULAANDTHEWALLISFORMULAISTWOSIMPLEPROMOTION,ASWELLASTHESERIESCONVERGENCEJUDGMENTAPPLICATIONASPECTSOFWALLISFORMULAFROMTHESEQUENCELIMITCALCULATION,INTEGRALCALCULATION,TOPROVIDESIGNIFICANTMATERIALANDIDEASFORTHETEACHINGOFCALCULUSKEYWORDSWALLISFORMULA,LIMIT,INTEGRAL目录引言11沃利斯公式的证明及推广111沃利斯公式的新证明1111有限次代数方程根与系数的关系类比到无限次方程1112应用含参量积分证明沃利斯公式312沃利斯公式的推广4121含参数的沃利斯公式4122含沃利斯公式的不等式52沃利斯公式的应用621沃利斯公式在极限计算中的应用622沃利斯公式在积分计算中的应用923沃利斯公式在级数收敛判别中的应用113总结13参考文献14引言近几年来，国内很多数学分析教材都引入WALLIS公式，关于其证明方法有很多种，一般都是利用积分证明的，本文将借助类比思维，分别利用SINDCOSNJXX根与系数关系的思维方法和含参量定积分来证明WALLIS公式此外，教材中关于其应用论述的很少，这是为什么呢因为很多可以应用WALLIS公式的“高地”被斯特林公式占领了但本文搜集到一些不能应用斯特林公式却可以能应用WALLIS公式的例子且WALLIS公式在推导斯特林公式中扮演很重要的角色，从加深理解WALLIS公式的角度探求其一些简单推广以及其在极限计算、积分计算和级数收敛判别方面的应用1沃利斯公式的证明及推广11沃利斯公式的新证明沃利斯公式指的是12124LIM31NN经过开平方后，则WALLIS公式可以写为2LINN现引入这样的数学记号，则WALLIS1351462N公式又可以写成（121LIMLIM22NNN或1）111有限次代数方程根与系数的关系类比到无限次方程类比的思维是人们把个别问题解决后所得到的经验用来解决其他近似问题的一种类似联想的思维的方法，类比这个重要的数学思想方法，曾被波利亚称为科学发现的“伟大引路人”，被17世纪德国著名天文学家和数学家开普勒视为“知道大自然一2切秘密”的“导师”在这我们也将采用类比思维对于有限次代数方程，假如有个不同的根2010NBXBXN1,K，那么左边的多项式就可以表示为线性因子乘积，即2,K3,NK2011231NNXXBXBK比较这个恒等式两边的同次幂的系数，就可以得到根和系数的关系特别是偶数次方程有个不相同的根242010NAXAX，则有12,N，我们比较42012NAX22220131NX二次项系数有10221NA根据幂级数展开式，在，则1X2468SIN3579XX利用无穷多项方程（12468103579XX2）由于方程（12）的根为，则,2,4,6246822221SIN111135793NXXXXXX即（121SINXN3）因为绝对收敛，所以这无穷乘积是绝对收敛的21NX在（13）中令，得,222111LIMLIM21TTTNNTT沃利斯公式（11）得证112应用含参量积分证明沃利斯公式引理1设，则有320,SINCOD,MNJX11,22,MJJN定理1设，证明120NIXNII证明令，根据引理1得SIT222001NCOD,2,SINCODNIJNJTTSITX令2112200DNNNXXXI由于，120D4IX12013I因此当时，即2,NM213121264MMI当时，210,N211232753MIM则120,2D,3NIXM,1N另一方面，由定积分的保不等式性质知，当时，有0,1X,112222212000DDDMMMXXX从而得到，21231MM从上式可得到22122311在上式中，令，则2MAM1223MA由于，因此根据迫敛性可知，因而2LILI13MM1LIMALI2MA2LI1WALLIS公式（11）得证12沃利斯公式的推广121含参数的沃利斯公式对任意非负实数和正整数，则有XN（12241LIM13NXXNX1XI4）其中20SIDXXIT4证明由分部积分法知，当时，则有2U100SINDSINDCOSUITT21UUII因此有21UUII于是，2312NXXNII，112X从而，21210NXNXXII即，2121NXNXII令，利用夹逼定理并整理得到（14）式N注1令，可以得到著名的WALLIS公式0X122含沃利斯公式的不等式关于WALLIS公式的研究一直以来都受数学家的关注，1956年21N5给出了如下含WALLIS公式形式的不等式KAZRINOF612114NN本文将含WALLIS公式不等式推广为当时，有下列式子成立2K,（11211KKKNKN5）或（12114KKNKNK6）证明如果，式（15）显然成立1如果，用数学归纳法证明，式（15）左边2K当时，显然成立N假设对式子（15）的左边对于正整数N成立，则下面证明对于同样成立，由1N归纳假设，只要证明,11KKKK即证明,KNN亦即（111KK7）根据伯努利不等式71KXK1,0XK或令，则XNK111KNNK所以式（17）成立因此，对任意正整数，式子（15）的左边成立下面证明式（15）的右边成立当时，要证明15的右边成立，只要证明1N,12KK即可，化简可知这个不等式成立的充要条件为，又由于时，有2K10KK因此，此时式（15）的右边成立假设式（15）的右边对于正整数，下面证明同样成立，只要证明NN,111NKKKK而此不等式成立的充要条件为,KKNNN即（111111KKNKNKK8）但是，由NEWTON二项式公式，式子（18）的右边不小于下面的式子121112KKKNNNN11KKKKKK1111KKKNNNKK所以式子（18）成立因此，对任意正整数,式子（15）的右边成立N2沃利斯公式的应用21沃利斯公式在极限计算中的应用由于沃利斯公式和极限有关，所以在计算一些极限的问题可以通过沃利斯公式会很容易出来例1求极限13521LIM46NN解利用沃利斯公式（13），可得21LI46NN21LIMNLIM221N1LILINN20例2设（），试证1NAN，LIM2NA2LINA解由于，21122NANN，21N因此，是递增数列根据沃利斯公式，则2NA21N，2LIMNA21LINA得证例3求极限2684LIM573NN解由沃利斯（WALLIS）公式的推广（14），则有221LI1NXNXX201SIDNXT令则4X224LIM131NXX268LI57NN4205SIDLIMNT9LI128N例4求极限21LI5NN解因为1221ARCSIXX2461213X462315XX，1NN,因此201ARCSINNXXD（2211NN1）由于当时，级数在处收敛（本文下面给予证明），又由X121NNX8于函数项级数M检验法知，级数1在上一致收敛,在（21）中，令，有SI2XTT，（2211SININTT2）对（22）所在区间取积分，并且由逐项积分公式，则有0,2，2100012SINDSINDNTTT，22101I8NNT又由沃利斯公式可知，20SIDT于是21281NN2200NN即211LIM9258N22沃利斯公式在积分计算中的应用对于一些不易用积分法求出原函数的积分，但是利用沃利斯公式却可能很容易解决这些问题例5求积分920XIED解假设，由X,246222421113XXXEXX可知,221X注意，前者仅对正确，而后者对任一都对，由此可得0X0，22NXE1221XN取积分22112200001NNXNXNDXDE但用替换可得UNX20DNXEI又,122100462SIN351NNXDT即,2200SI142NNTDXN所以4613352IN平方得22224131135144NNNIN由沃利斯公式得224LIM1351NN可知，当时N,2I即24I因此20DXE例6求的值1010SINDJX解10210I97538642150例7求积分的值35214DXX解335522119DX令,则2SINXCOS原式3223INSICOSD32SI7C4281INOSD44226C3CSIN4201OSD323沃利斯公式在级数收敛判别中的应用对于一些级数收敛性的判别问题，文献10指出若利用沃利斯公式可能会起到事半功倍的效果例8判别正项级数，（）的敛散性12SNR证由于通项含有双阶乘的运算，原则上想到运用比式判别法，但21SNU是由于，因此比式判别法失效1LIMN若运用拉贝判别法，由于12LILIM1SNNNA，LI2NSS所以当时，即时收敛，当时则发散，但当时拉贝判别法则无法进12S2S2P行判别但如果利用沃利斯公式，不仅对于和时的情况可以判别，而且对时2S2S的情况也能判别比如由沃利斯公式得122112NNN则正项级数与正项级数的敛散性相同由上分析可得正项级数12SN12SN当时收敛，当时发散12SN例9二项式级数111MNNMXX当，时条件收敛1X0证令，表示二项式级数在时的通项，则UNX，11,23NNN故此二项式级数是一交错级数，且，1NU0,由于，则必存在两个正整数和，使01MKJ，11KMJ再结合沃利斯公式的推论中式子（15）可得NNU121KKKN即，由判别法可知级数收敛LIM0NULEIBZ0NU又11NJJUN沃利斯公式推广中公式16得112NJJ1N214JNJN再由调和级数发散可知级数发散1N1NU所以当，时二项式级数条件收敛X0M3总结本文针对沃利斯公式的应用进行研究，给出了沃利斯公式在求某些极限计算、积分计算、级数收敛的简便之处并且将沃利斯公式进行简单的推广，在证明某些级数收敛性问题时，运用达朗贝尔法与拉贝判别法时会失效，但运用沃利斯公式会很简单有效的解决这类问题，此外我们知道在有关二项式级数在收敛区间端点的收敛性是一个较为困难的问题，有的教材对此置之不理，有的则要借助于几何级数来解决，本文利用对沃利斯公式的推广能有效的解决一些此类型的问题当然还有更多问题值得我们探讨，例如对含参数的沃利斯公式的更多应用以及含沃利斯公式的双边不等式的推广可以给出更为精确的结果，以及沃利斯公式在二项式的上下界的研究等，这些问题将2NC另文研究参考文献1华东师范大学数学系数学分析上册（第四版）北京高等教育出版社，20012屈芝莲WALLIS公式新证明科学技术与工程,2011,15495503MIKHAILKOVALYOVELEMENTARYCOMBINATORIALPROBABILISTICPROOFOFTHEWALLISFORMULASJOURNALOFMATHEMATICSANDSTATISTICS,2009,54084094李建军一种含参数的WALLIS公式与STIRLING公式数学理论与应用,2008,352535赵德钧关于含WALLIS公式的双边不等式数学的实践与认识，2004,3471671686DKKAZARINOFFONWALLISFORMULAEDINBURGHMATHNOTES,1956,4019217张文亮一个不等式的探讨2004,319218JEFFREYHWALLISFORMULAFORMETHODSOFMATHEMATICALPHYSICS,1988,34684679王振芳,陈慧琴沃利斯WALLIS公式及其应用山西大同大学学报,2011,105610张国铭WALLIS公式的几个应用高等数学研究,2008,93740请您删除一下内容，O_O谢谢MANYPEOPLEHAVETHESAMEMIXEDFEELINGSWHENPLANNINGATRIPDURINGGOLDENWEEKWITHHEAPSOFTIME,THESEVENDAYCHINESE请您删除一下内容，O_O谢谢NATIONALDAYHOLIDAYCOULDBETHEBESTOCCASIONTOENJOYADESTINATIONHOWEVER,ITCANALSOBETHEEASIESTWAYTORUINHOWYOUFEELABOUTAPLACEANDYOUMAYBECOMEMOREFATIGUEDAFTERTHEHOLIDAY,DUETOBATTLINGTHELARGECROWDSDURINGPEAKSEASON,ADREAMABOUTAPLACECANTURNTONIGHTMAREWITHOUTCAREFULPLANNING,ESPECIALLYIFYOUTRAVELWITHCHILDRENANDOLDERPEOPLEASMOSTCHINESEPEOPLEWILLTAKETHEHOLIDAYTOVISITDOMESTICTOURISTDESTINATIONS,CROWDSANDBUSYTRAFFICAREINEVITABLEATMOSTPLACESALSOTOBEEXPECTEDAREINCREASINGTRANSPORTANDACCOMMODATIONPRICES,WITHTHEPOSSIBILITYTHATTHEREWILLBENOROOMSAVAILABLEITISALSOCOMMONTHATYOULLWAITINTHELINEFORONEHOURTOGETATICKET,ANDANOTHERTWOHOURSATTHESITE,TOONLYSEEATINYBITOFTHEPLACEDUETOTHECROWDSLASTYEAR,428MILLIONTOURISTSTRAVELEDINCHINAOVERTHEWEEKLONGHOLIDAYINOCTOBERTRAVELINGDURINGTHISPERIODISAMATTERTHATNEEDSTHOROUGHPREPARATIONIFYOUARESHORTONTIMETOPLANTHEUPCOMING“GOLDENWEEK“ITMAYNOTBEABADIDEATOAVOIDSOMEOFTHEMOSTCROWDEDPLACESFORNOWTHEREISALWAYSAPLACESOFASCINATINGTHATEVERYONEYEARNSFORARXANISAPLACELIKETHISTHEBEAUTYOFARXANISEVERLASTINGREGARDLESSOFTHECHANGINGOFFOURSEASONSBESTOWEDBYNATURE,ITSSPECTACULARSEASONALLANDSCAPEANDMOUNTAINSAREJUSTBEYONDWORDARXANISACRUCIALDESTINATIONFORTHERECOMMENDEDTRAVELLINGROUTE,“CHINAINNERMONGOLIAARXANHAILARMANZHOULI“ITISALSOTHEJOINTOFTHEFOURPRAIRIESACROSSTHESINOMONGOLIANBORDER,WHEREPEOPLEGRAVITATETOWARDSTHEEXOTICATMOSPHEREMIXEDWITHCHINESE,RUSSIAN,ANDMONGOLIAELEMENTSASAHISTORICSITEFORTHEYITIANBATTLE,ARXANSTILLEMBODIESTHESPIRITOFGENGHISKHANWALKINGINTOARXAN,YOUWILLBEAMAZEDBYAKALEIDOSCOPEOFGORGEOUSCOLORSALLTHEYEARROUNDTHESPRINGAZALEASBLOOMINGREDINTHESNOW,THESUMMERSEAWAVERINGBLUEINTHEBREEZE,THEAUTUMNLEAVESPAINTEDINYELLOWCOVERINGVOLCANICTRACES,ANDTHEWINTERWOODSSHININGWHITEONTHEVASTALPINESNOWSCAPEHINGGANLEAGUEARXANCITYISSITUATEDINTHEFAREASTERNAREAOFINNERMONGOLIAAUTONOMOUSREGIONITSFULLNAME“HARENARXAN“MEANS“HOTHOLYWATER“INTHEMONGOLIANLANGUAGEARXANISATOURISMCITYINTHENORTHERNFRONTIERWITHABLENDOFLARGEFOREST,GRANDPRAIRIES,VASTSNOWFIELD,HEAVENLAKECLUSTER,THERMIUM,ASWELLASVOLCANICCLUSTERITISARAREANDUNIQUEECOTOURISMBASEFILLEDWITHHEALTHYSUNSHINE,CLEANAIRANDUNSPOILEDGREENNESTLEDCLOSETOTHECOUNTRYSLARGESTVIRGINFOREST,ANDKNOWNFORITSSPRINGANDECOLOGICALENVIRONMENT,ARXANISMARVELEDATBYMANYTOURISTSASTHEPURESTLANDONEARTHYOUCANNOTMISSOUTTHEAUTUMNOFARXANITISDEFINITELYTHEBESTWITHBRIGHTLYCOLOREDSCENERYFULLOFEMOTIONSAUTUMNINTHENORTHERNPARTOFTHECOUNTRYCOMESEARLIERTHANTHESOUTHASEPTEMBERRAINFOLLOWEDBYTHEFOOTPRINTSOFAUTUMNBRINGSMORECOLORSTOTHEONCEEMERALDGREENMOUNTAINANDBLOOMINGGRASSLANDSHUTTERBUGSFLOCKTOSEEFORTHEMSELVESTHEMARVELOFSPLENDIDCOLORSAROUNDTHEMOUNTAINSANDWATERS,MANYOFWHOMHAVETRAVELALONGDISTANCEANDEVENCAMPHEREONLYTOCAPTUREAMOMENTOFTHENATUREWONDERTHESILVERBIRCHTURNSGOLDEN,WHILETHELARCHISSTILLPROUDLYGREENYOUWILLFINDYOURSELFDROWNEDINTHEINTOXICATINGREDOFTHEWILDFRUITSASWELLASTHEGLAMOUROFFLOWERSINFULLBLOWNANDYOURHEARTWILLBELINGERINGONTHEWOODSASITSTIMEFORTHEWILDFRUITSTORIPETHEPICTURESQUEARXANINAUTUMNISINDEEDAFAIRYLANDONLYEXISTSINADREAMTHATSATISFIESALLYOURFANTASIESIFITRAINSHEAVILYONSATURDAYNIGHT,SOMEELDERLYCHINESEWILLSAYITISBECAUSEZHINU,ORTHEWEAVINGMAID,ISCRYINGONTHEDAYSHEMETHERHUSBANDNIULANG,ORTHECOWHERD,ONTHEMILKYWAYMOSTCHINESEREMEMBERBEINGTOLDTHISROMANTICTRAGEDYWHENTHEYWERECHILDRENONQIXI,ORTHESEVENTHNIGHTFESTIVAL,WHICHFALLSONTHESEVENTHDAYOFTHESEVENTHLUNARMONTH,WHICHISUSUALLYINEARLYAUGUSTTHISYEARITFALLSONSATURDAY,AUGUST2FOLKLORESTORYASTHESTORYGOES,ONCETHEREWASACOWHERD,NIULANG,WHOLIVEDWITHHISELDERBROTHERANDSISTERINLAWBUTSHEDISLIKEDANDABUSEDHIM,ANDTHEBOYWASFORCEDTOLEAVEHOMEWITHONLYANOLDCOWFORCOMPANYTHECOW,HOWEVER,WASAFORMERGODWHOHADVIOLATEDIMPERIALRULESANDWASSENTTOEARTHINBOVINEFORMONEDAYTHECOWLEDNIULANGTOALAKEWHEREFAIRIESTOOKABATHONEARTHAMONGTHEMWASZHINU,THEMOSTBEAUTIFULFAIRYANDASKILLEDSEAMSTRESSTHETWOFELLINLOVEATFIRSTSIGHTANDWERESOONMARRIEDTHEYHADASONANDDAUGHTERANDTHEIRHAPPYLIFEWASHELDUPASANEXAMPLEFORHUNDREDSOFYEARSINCHINAYETINTHEEYESOFTHEJADEEMPEROR,THESUPREMEDEITYINTAOISM,MARRIAGEBETWEENAMORTALANDFAIRYWASSTRICTLYFORBIDDENHEORDEREDTHEHEAVENTROOPTOCATCHZHINUBACKNIULANGGREWDESPERATEWHENHEDISCOVEREDZHINUHADBEENTAKENBACKTOHEAVENDRIVENBYNIULANGSMISERY,THECOWTOLDHIMTOTURNITSHIDEINTOAPAIROFSHOESAFTERITDIEDTHEMAGICSHOESWHISKEDNIULANG,WHOCARRIEDHISTWOCHILDRENINBASKETSSTRUNGFROMASHOULDERPOLE,OFFONACHASEAFTERTHEEMPRESSTHEPURSUITENRAGEDTHEEMPRESS,WHOTOOKHERHAIRPINANDSLASHEDITACROSSTHESKYCREATINGTHEMILKYWAYWHICHSEPARATEDHUSBANDFROMWIFEBUTALLWASNOTLOSTASMAGPIES,MOVEDBYTHEIRLOVEANDDEVOTION,FORMEDABRIDGEACROSSTHEMILKYWAYTOREUNITETHEFAMILYEVENTHE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