數(shù)學(xué)專業(yè)畢業(yè)論文_正項(xiàng)級數(shù)斂散性判別法的比較及其應(yīng)用

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11、aining正項(xiàng)級數(shù)判別法的知道及其應(yīng)用畢業(yè)論文目 錄摘要1關(guān)鍵詞1Abstract1Key words1引言11正項(xiàng)級數(shù)相關(guān)概念21.1正項(xiàng)級數(shù)的定義21.2正項(xiàng)級數(shù)斂散性判別的充要條件22正項(xiàng)級數(shù)斂散性判別法22.1判別級數(shù)發(fā)散的簡單方法22.2比較判別法32.2.1定理及其極限形式32.2.2活用比較判別法32.3柯西判別法42.3.1定理及其極限形式42.3.2 活用柯西判別法52.4達(dá)朗貝爾判別法52.4.1定理及其極限形式52.4.2活用達(dá)朗貝爾判別法62.5積分判別法62.5.1定理625.2活用積分判別法62.6拉貝判別法62.6.1定理及其極限形式72.6.2活用拉貝判別法7

12、2.7其他判別法83判別方法的比較 93.1不同方法的比較及應(yīng)用103.2判別正項(xiàng)級數(shù)斂散性方法的總結(jié)11致謝12參考文獻(xiàn)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正項(xiàng)級數(shù)斂散性判別法的比較及其應(yīng)用數(shù)學(xué)與應(yīng)用數(shù)學(xué) xx指導(dǎo)教師 xx摘要：正項(xiàng)級數(shù)是級數(shù)內(nèi)容中的一種重要級數(shù),它的斂散性是其基本性質(zhì).正項(xiàng)級數(shù)斂散性判斷的方法雖然較多，但使用起來仍有一定的技巧，歸納總結(jié)正項(xiàng)級數(shù)收斂性判斷的一些典型方法，比較這些方法的不同特點(diǎn)，總結(jié)

23、出一些典型的正項(xiàng)級數(shù)，根據(jù)不同的題目特點(diǎn)分析、判斷選擇適宜的方法進(jìn)行判斷，才能事半功倍.關(guān)鍵詞：正項(xiàng)級數(shù)；收斂；方法；比較；應(yīng)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 我們在書上已經(jīng)學(xué)了很多種正項(xiàng)級數(shù)斂散性的判定定理,但書上往往只是對定理本身做一個證明，然后舉幾個簡單應(yīng)用的例子就好了，沒有做

26、過多的分析.但是,我們在實(shí)際做題目時，常會有這些感覺：有時不知該選用哪種方法比較好；有時用這種或那種方法時，根本做不出來，也就是說，定理它本身存在著一些局限性.因此，我們便會去想，我們常用的這些定理到底有哪些局限呢，定理與定理之間會有些什么聯(lián)系和區(qū)別呢，做題目時如何才能更好得去運(yùn)用這些定理呢? 這就是本文所要討論的.1 正項(xiàng)級數(shù)相關(guān)概念1.1正項(xiàng)級數(shù)的定義 如果級數(shù)的各項(xiàng)都是非負(fù)實(shí)數(shù)，即則稱此級數(shù)為正項(xiàng)級數(shù)1.2正項(xiàng)級數(shù)斂散性判別的充要條件正項(xiàng)級數(shù)的每一項(xiàng)都為正的基本特點(diǎn)導(dǎo)致正項(xiàng)級數(shù)部分和數(shù)列單調(diào)增加,從而有正項(xiàng)級數(shù)斂散性的基本判別定理:定理1 正項(xiàng)級數(shù)收斂它的部分和數(shù)列有上界.證明 由于,所

27、以是遞增數(shù)列.而單調(diào)數(shù)列收斂的充要條件是該數(shù)列有界(單調(diào)有界定理),從而本定理得證.例 級數(shù)是正項(xiàng)級數(shù)。它的部分和數(shù)列的通項(xiàng) ，所以正項(xiàng)級數(shù)收斂。2 正項(xiàng)級數(shù)斂散性判別法21 判別發(fā)散的簡單方法由級數(shù)收斂的基本判別定理柯西收斂準(zhǔn)則:級數(shù)收斂有.取特殊的,可得推論:若級數(shù)收斂,則.定理2 該推論的逆否命題:若,則級數(shù)發(fā)散.例1 快速判斷級數(shù)的斂散性. 解: 由于,從而根據(jù)定理2可知,該級數(shù)發(fā)散.如果,則可由該逆否命題直接可以判別出該級數(shù)發(fā)散;如果,則不能判斷級數(shù)是否收斂,因?yàn)榇嬖诩墧?shù)滿足的發(fā)散級數(shù)，如；也存在級數(shù)滿足的收斂級數(shù)，如.顯然該逆否命題只使用于滿足的發(fā)散級數(shù).22 比較判別法221 定

28、理及其極限形式 定理3 (比較判別法) 有兩個正項(xiàng)級數(shù)與,且,有,c是正常數(shù). 1）若級數(shù)收斂,則級數(shù)也收斂; 2）若級數(shù)發(fā)散,則級數(shù)也發(fā)散.比較判別法的極限形式 有兩個正項(xiàng)級數(shù)與,且 1）若級數(shù)收斂,且,則級數(shù)也收斂; 2）若級數(shù)發(fā)散,且,則級數(shù)也發(fā)散.222 活用比較判別法 (1) 當(dāng)所求級數(shù)的通項(xiàng)中出現(xiàn)關(guān)于n的有理式時,比較對象常常選取p-級數(shù)或調(diào)和級數(shù). 例1 判別級數(shù)的斂散性.解: 因?yàn)椋ǚ帜缚s小,分?jǐn)?shù)放大）,又由于收斂.則由此比較判別法,原級數(shù)也收斂.(2) 當(dāng)所求級數(shù)通項(xiàng)中出現(xiàn)正弦函數(shù)或?qū)?shù)函數(shù)時,利用不等式選取適當(dāng)?shù)谋容^對象. 例2 判別級數(shù)的斂散性.分析: 考慮當(dāng)時,則,而是

29、公比的收斂級數(shù),故原級數(shù)收斂.(3) 當(dāng)所求級數(shù)的通項(xiàng)放大、縮小不方便時,可采用比較判別法的推論.例3 判別級數(shù)的斂散性.解: 由于,又因?yàn)槭前l(fā)散的,則原級數(shù)也發(fā)散.23 柯西判別法(根值判別法)231 定理及其極限形式 定理4 (柯西判別法) 有正項(xiàng)級數(shù),存在常數(shù). 1）若,有 ,則級數(shù)收斂; 2）若存在無限個n,有 ,則級數(shù)發(fā)散.證明 1）已知有 或 .又已知幾何級數(shù)收斂,于是級數(shù)收斂.2）已知存在無限個n,有 或 ,即不趨近于,于是級數(shù)發(fā)散.根值判別法的極限形式 有正項(xiàng)級數(shù),若 ,則 1）當(dāng)時,級數(shù)收斂; 2）當(dāng)時,級數(shù)發(fā)散.232 活用柯西判別法 例1 判別級數(shù)的斂散性.解: 由于,根

30、據(jù)柯西判別法的推論,可得級數(shù)收斂.例2 判別級數(shù)的斂散性。解: 由于,所以根據(jù)柯西判別法的推論知,級數(shù)發(fā)散.24 達(dá)朗貝爾判別法(比值判別法)241 定理及其極限形式 定理5 (達(dá)朗貝爾判別法) 有正項(xiàng)級數(shù),存在常數(shù). 1）若,有 ,則級數(shù)收斂; 2）若,有 ,則級數(shù)發(fā)散.比值判別法的極限形式 有正項(xiàng)級數(shù),且1）當(dāng)時,級數(shù)收斂;2）當(dāng)時,級數(shù)發(fā)散.242 活用達(dá)朗貝爾判別法 例1 判別級數(shù)的斂散性.解: 由于,所以根據(jù)達(dá)朗貝爾判別法的推論知,級數(shù)收斂.例2 判別級數(shù)的斂散性.解: 由于,根據(jù)達(dá)朗貝爾判別法的推論知,級數(shù)發(fā)散.25 積分判別法251 定理 (積分判別法) 設(shè)為上非負(fù)減函數(shù),那么正

31、項(xiàng)級數(shù)與反常積分同時收斂或同時發(fā)散. 252 活用積分判別法 例1 判別級數(shù)的斂散性.解:將原級數(shù)換成積分形式,由于,即收斂,根據(jù)積分判別法可知,級數(shù)也收斂.例2 證明調(diào)和級數(shù)發(fā)散.解:將原級數(shù)換成積分形式,由于,即發(fā)散,根據(jù)積分判別法可知,調(diào)和級數(shù)發(fā)散.26 拉貝判別法261 定理及其極限形式 (拉貝判別法) 有正項(xiàng)級數(shù),存在常數(shù).1) 若,有,則級數(shù)收斂；2) 若,有,則級數(shù)發(fā)散.拉貝判別法極限形式 有正項(xiàng)級數(shù),且極限存在,若1) 當(dāng)時,級數(shù)收斂;2) 當(dāng)時,級數(shù)發(fā)散.262 活用拉貝判別法 例1 討論級數(shù)當(dāng)時的斂散性.解: 當(dāng)時,由于,所以根據(jù)拉貝判別法知,原級數(shù)是發(fā)散的.當(dāng)時,由于,所

32、以原級數(shù)是發(fā)散的.當(dāng)時,由于,所以原級數(shù)收斂.27其它判別法 （一）阿貝爾判別法若數(shù)列，且為單調(diào)有界數(shù)列，級數(shù)收斂，則級數(shù)收斂。 （二）狄利克雷判別法若數(shù)列，且數(shù)列單調(diào)遞減，又級數(shù)的部分和數(shù)列有界，則級數(shù)收斂。（三）伯爾特昂（Bertrand）判別法設(shè)是正項(xiàng)級數(shù)，且，若，則（1）當(dāng)B1時，級數(shù)收斂；（2）當(dāng)BN，成立不等式，則級數(shù)收斂；（2） 對所有nN，成立不等式，則級數(shù)發(fā)散。庫默爾判別法的極限形式 設(shè)是正項(xiàng)級數(shù)，是使級數(shù)發(fā)散的正數(shù)列，設(shè)，（有限或無限）（1）當(dāng)時，級數(shù)收斂；（2）當(dāng)時，級數(shù)發(fā)散。注：使用庫默爾判別法時，應(yīng)注意選擇恰當(dāng)?shù)?，?dāng)時，可以推導(dǎo)出比式判別法；當(dāng)時，可以推導(dǎo)出拉貝判別法

33、；當(dāng)時，可以推導(dǎo)出伯爾特昂（Bertrand）判別法。（五）高斯判別法 設(shè)是正項(xiàng)級數(shù)，若，為常數(shù)，為有界量，又，則（1）當(dāng)時，級數(shù)收斂；（2）當(dāng)時，級數(shù)發(fā)散。（六）厄耳瑪可夫判別法 設(shè)為單調(diào)遞減的正值函數(shù)，若，則（1）當(dāng)時，級數(shù)收斂；（2）當(dāng)時，級數(shù)發(fā)散。3 判別方法的比較31 不同方法的比較及應(yīng)用 （一）當(dāng)級數(shù)可化為含參數(shù)的一般式、通項(xiàng)為等差或等比式或通項(xiàng)為含二項(xiàng)以上根式的四則運(yùn)算且通項(xiàng)極限無法求出時，可以選用正項(xiàng)級數(shù)的充要條件進(jìn)行判斷。如：（二）當(dāng)級數(shù)表達(dá)式型如，為任意函數(shù)、級數(shù)一般項(xiàng)如含有或等三角函數(shù)的因子可以進(jìn)行適當(dāng)?shù)姆趴s，并與幾何級數(shù)、P級數(shù)、調(diào)和級數(shù)進(jìn)行比較、不易算出或、等此類無法

34、判斷級數(shù)收斂性或進(jìn)行有關(guān)級數(shù)的證明問題時，應(yīng)選用比較判別法。例：（1） 級數(shù)收斂（2） 級數(shù)收斂比較判別法使用的范圍比較廣泛，適用于大部分無法通過其它途徑判別其斂散性的正項(xiàng)級數(shù)。（三）當(dāng)級數(shù)含有階層、n次冪，型如或或分子、分母含多個因子連乘除時，選用比式判別法。當(dāng)通項(xiàng)含與的函數(shù)可以選用比式判別法的極限形式進(jìn)行判斷，例：（1）所以級數(shù)收斂（2）級數(shù)收斂一般來說，當(dāng)選用根式判別法無法判斷時，我們也可以選用比式判別法來判斷，但有時候我們用根式判別法而不使用比式判別法，因?yàn)楦脚袆e法得到的收斂條件比比式判別法更優(yōu)。（五）當(dāng)級數(shù)表達(dá)式型如，為含有的表達(dá)式或可以找到原函數(shù)，或級數(shù)為上非負(fù)單調(diào)遞減函數(shù)，含有

35、等三角函數(shù)的因子可以找到原函數(shù)，可以選用積分判別法。例：（六）當(dāng)級數(shù)同時含有階層與n次冪，型如與時，或使用比值、根式判別法時極限等于1或無窮無法判斷其斂散性的時候，選用拉貝判別法。例：不能用比值判別法無法判斷斂散性不能用根式判別法 無法判斷斂散性因此，當(dāng)根式判別法與比值判別法無法判斷斂散性時，我們可以選用拉貝判別法。（七）當(dāng)通項(xiàng)是由兩個部分乘積而成，其中一部分為單調(diào)遞減且極限趨于0的數(shù)列，另一部分為部分和有界的數(shù)列，如含有等三角函數(shù)、等；或可化為，如；也可以型如，為任意函數(shù)，則可以選用狄利克雷判別法。阿貝爾判別法也可以看成是狄利克雷判別法的特殊形式。例：（八）當(dāng)通項(xiàng)型如，其中為含的函數(shù)，可以選擇伯爾特昂判別法。如：（九）當(dāng)?shù)闹悼苫?/p>