数学专业毕业论文_正项级数敛散性判别法的比较及其应用

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11、aining正项级数判别法的知道及其应用毕业论文目 录摘要1关键词1Abstract1Key words1引言11正项级数相关概念21.1正项级数的定义21.2正项级数敛散性判别的充要条件22正项级数敛散性判别法22.1判别级数发散的简单方法22.2比较判别法32.2.1定理及其极限形式32.2.2活用比较判别法32.3柯西判别法42.3.1定理及其极限形式42.3.2 活用柯西判别法52.4达朗贝尔判别法52.4.1定理及其极限形式52.4.2活用达朗贝尔判别法62.5积分判别法62.5.1定理625.2活用积分判别法62.6拉贝判别法62.6.1定理及其极限形式72.6.2活用拉贝判别法7

12、2.7其他判别法83判别方法的比较 93.1不同方法的比较及应用103.2判别正项级数敛散性方法的总结11致谢12参考文献12supervision in large and medium goods vehicle. A is established large vehicles and small vehicles classification management of motor vehicle test mode, increased medium van car, and dangerous goods transport car, and school car test proj

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22、ation Highway, and continuous urgent detours, and snow days, and wet sliding road, and burst situation disposal, subjects II field to driving exam project, is expected2正项级数敛散性判别法的比较及其应用数学与应用数学 xx指导教师 xx摘要：正项级数是级数内容中的一种重要级数,它的敛散性是其基本性质.正项级数敛散性判断的方法虽然较多，但使用起来仍有一定的技巧，归纳总结正项级数收敛性判断的一些典型方法，比较这些方法的不同特点，总结

23、出一些典型的正项级数，根据不同的题目特点分析、判断选择适宜的方法进行判断，才能事半功倍.关键词：正项级数；收敛；方法；比较；应用Positive Series Convergence Criterion of Comparison and Its ApplicationMathematics and Applied Mathematics LiQinglin Tutor LiPingrunAbstract：Positive series is a series of important theoretical component and its convergence is the core

24、issue of series theory .Although positive series convergence judgment methods more ,there still have to use the skills, summarized convergence of positive series to determine some of the typical method to compare the different characteristics of these methods, summed up the typical positive series,

25、according to the characteristics of different subject analysis to determine to choose suitable methods to judge, to maximize savings in time and increase efficiency.Key words: positive series ; convergence; methods; compare；application引言 我们在书上已经学了很多种正项级数敛散性的判定定理,但书上往往只是对定理本身做一个证明，然后举几个简单应用的例子就好了，没有做

26、过多的分析.但是,我们在实际做题目时，常会有这些感觉：有时不知该选用哪种方法比较好；有时用这种或那种方法时，根本做不出来，也就是说，定理它本身存在着一些局限性.因此，我们便会去想，我们常用的这些定理到底有哪些局限呢，定理与定理之间会有些什么联系和区别呢，做题目时如何才能更好得去运用这些定理呢? 这就是本文所要讨论的.1 正项级数相关概念1.1正项级数的定义 如果级数的各项都是非负实数，即则称此级数为正项级数1.2正项级数敛散性判别的充要条件正项级数的每一项都为正的基本特点导致正项级数部分和数列单调增加,从而有正项级数敛散性的基本判别定理:定理1 正项级数收敛它的部分和数列有上界.证明 由于,所

27、以是递增数列.而单调数列收敛的充要条件是该数列有界(单调有界定理),从而本定理得证.例 级数是正项级数。它的部分和数列的通项 ，所以正项级数收敛。2 正项级数敛散性判别法21 判别发散的简单方法由级数收敛的基本判别定理柯西收敛准则:级数收敛有.取特殊的,可得推论:若级数收敛,则.定理2 该推论的逆否命题:若,则级数发散.例1 快速判断级数的敛散性. 解: 由于,从而根据定理2可知,该级数发散.如果,则可由该逆否命题直接可以判别出该级数发散;如果,则不能判断级数是否收敛,因为存在级数满足的发散级数，如；也存在级数满足的收敛级数，如.显然该逆否命题只使用于满足的发散级数.22 比较判别法221 定

28、理及其极限形式 定理3 (比较判别法) 有两个正项级数与,且,有,c是正常数. 1）若级数收敛,则级数也收敛; 2）若级数发散,则级数也发散.比较判别法的极限形式 有两个正项级数与,且 1）若级数收敛,且,则级数也收敛; 2）若级数发散,且,则级数也发散.222 活用比较判别法 (1) 当所求级数的通项中出现关于n的有理式时,比较对象常常选取p-级数或调和级数. 例1 判别级数的敛散性.解: 因为（分母缩小,分数放大）,又由于收敛.则由此比较判别法,原级数也收敛.(2) 当所求级数通项中出现正弦函数或对数函数时,利用不等式选取适当的比较对象. 例2 判别级数的敛散性.分析: 考虑当时,则,而是

29、公比的收敛级数,故原级数收敛.(3) 当所求级数的通项放大、缩小不方便时,可采用比较判别法的推论.例3 判别级数的敛散性.解: 由于,又因为是发散的,则原级数也发散.23 柯西判别法(根值判别法)231 定理及其极限形式 定理4 (柯西判别法) 有正项级数,存在常数. 1）若,有 ,则级数收敛; 2）若存在无限个n,有 ,则级数发散.证明 1）已知有 或 .又已知几何级数收敛,于是级数收敛.2）已知存在无限个n,有 或 ,即不趋近于,于是级数发散.根值判别法的极限形式 有正项级数,若 ,则 1）当时,级数收敛; 2）当时,级数发散.232 活用柯西判别法 例1 判别级数的敛散性.解: 由于,根

30、据柯西判别法的推论,可得级数收敛.例2 判别级数的敛散性。解: 由于,所以根据柯西判别法的推论知,级数发散.24 达朗贝尔判别法(比值判别法)241 定理及其极限形式 定理5 (达朗贝尔判别法) 有正项级数,存在常数. 1）若,有 ,则级数收敛; 2）若,有 ,则级数发散.比值判别法的极限形式 有正项级数,且1）当时,级数收敛;2）当时,级数发散.242 活用达朗贝尔判别法 例1 判别级数的敛散性.解: 由于,所以根据达朗贝尔判别法的推论知,级数收敛.例2 判别级数的敛散性.解: 由于,根据达朗贝尔判别法的推论知,级数发散.25 积分判别法251 定理 (积分判别法) 设为上非负减函数,那么正

31、项级数与反常积分同时收敛或同时发散. 252 活用积分判别法 例1 判别级数的敛散性.解:将原级数换成积分形式,由于,即收敛,根据积分判别法可知,级数也收敛.例2 证明调和级数发散.解:将原级数换成积分形式,由于,即发散,根据积分判别法可知,调和级数发散.26 拉贝判别法261 定理及其极限形式 (拉贝判别法) 有正项级数,存在常数.1) 若,有,则级数收敛；2) 若,有,则级数发散.拉贝判别法极限形式 有正项级数,且极限存在,若1) 当时,级数收敛;2) 当时,级数发散.262 活用拉贝判别法 例1 讨论级数当时的敛散性.解: 当时,由于,所以根据拉贝判别法知,原级数是发散的.当时,由于,所

32、以原级数是发散的.当时,由于,所以原级数收敛.27其它判别法 （一）阿贝尔判别法若数列，且为单调有界数列，级数收敛，则级数收敛。 （二）狄利克雷判别法若数列，且数列单调递减，又级数的部分和数列有界，则级数收敛。（三）伯尔特昂（Bertrand）判别法设是正项级数，且，若，则（1）当B1时，级数收敛；（2）当BN，成立不等式，则级数收敛；（2） 对所有nN，成立不等式，则级数发散。库默尔判别法的极限形式 设是正项级数，是使级数发散的正数列，设，（有限或无限）（1）当时，级数收敛；（2）当时，级数发散。注：使用库默尔判别法时，应注意选择恰当的，当时，可以推导出比式判别法；当时，可以推导出拉贝判别法

33、；当时，可以推导出伯尔特昂（Bertrand）判别法。（五）高斯判别法 设是正项级数，若，为常数，为有界量，又，则（1）当时，级数收敛；（2）当时，级数发散。（六）厄耳玛可夫判别法 设为单调递减的正值函数，若，则（1）当时，级数收敛；（2）当时，级数发散。3 判别方法的比较31 不同方法的比较及应用 （一）当级数可化为含参数的一般式、通项为等差或等比式或通项为含二项以上根式的四则运算且通项极限无法求出时，可以选用正项级数的充要条件进行判断。如：（二）当级数表达式型如，为任意函数、级数一般项如含有或等三角函数的因子可以进行适当的放缩，并与几何级数、P级数、调和级数进行比较、不易算出或、等此类无法

34、判断级数收敛性或进行有关级数的证明问题时，应选用比较判别法。例：（1） 级数收敛（2） 级数收敛比较判别法使用的范围比较广泛，适用于大部分无法通过其它途径判别其敛散性的正项级数。（三）当级数含有阶层、n次幂，型如或或分子、分母含多个因子连乘除时，选用比式判别法。当通项含与的函数可以选用比式判别法的极限形式进行判断，例：（1）所以级数收敛（2）级数收敛一般来说，当选用根式判别法无法判断时，我们也可以选用比式判别法来判断，但有时候我们用根式判别法而不使用比式判别法，因为根式判别法得到的收敛条件比比式判别法更优。（五）当级数表达式型如，为含有的表达式或可以找到原函数，或级数为上非负单调递减函数，含有

35、等三角函数的因子可以找到原函数，可以选用积分判别法。例：（六）当级数同时含有阶层与n次幂，型如与时，或使用比值、根式判别法时极限等于1或无穷无法判断其敛散性的时候，选用拉贝判别法。例：不能用比值判别法无法判断敛散性不能用根式判别法 无法判断敛散性因此，当根式判别法与比值判别法无法判断敛散性时，我们可以选用拉贝判别法。（七）当通项是由两个部分乘积而成，其中一部分为单调递减且极限趋于0的数列，另一部分为部分和有界的数列，如含有等三角函数、等；或可化为，如；也可以型如，为任意函数，则可以选用狄利克雷判别法。阿贝尔判别法也可以看成是狄利克雷判别法的特殊形式。例：（八）当通项型如，其中为含的函数，可以选择伯尔特昂判别法。如：（九）当的值可化