3第一讲__数列的极限典型例题第一讲数列的极限一、内容提要1.数列极限的定义«Skip Record If...»，有«Skip Record If...».注1 «Skip Record If...»的双重性.一方面,正数«Skip Record If...»具有绝对的任意性,这样才能有«Skip Record If...»无限趋近于«Skip Record If...»另一方面,正数«Skip Record If...»又具有相对的固定性,从而使不等式«Skip Record If...».还表明数列«Skip Record If...»无限趋近于«Skip Record If...»的渐近过程的不同程度,进而能估算«Skip Record If...»趋近于«Skip Record If...»的近似程度. 注2若«Skip Record If...»存在，则对于每一个正数«Skip Record If...»，总存在一正整数«Skip Record If...»与之对应，但这种«Skip Record If...»不是唯一的，若«Skip Record If...»满足定义中的要求，则取«Skip Record If...»，作为定义中的新的一个«Skip Record If...»也必须满足极限定义中的要求，故若存在一个«Skip Record If...»则必存在无穷多个正整数可作为定义中的«Skip Record If...»．注3«Skip Record If...»«Skip Record If...»的几何意义是：对«Skip Record If...»的预先给定的任意«Skip Record If...»邻域«Skip Record If...»，在«Skip Record If...»中至多除去有限项，其余的无穷多项将全部进入«Skip Record If...»．注4«Skip Record If...»，有«Skip Record If...».2.子列的定义在数列«Skip Record If...»中，保持原来次序自左往右任意选取无穷多个项所得的数列称为«Skip Record If...»的子列，记为«Skip Record If...»，其中«Skip Record If...»表示«Skip Record If...»在原数列中的项数，«Skip Record If...»表示它在子列中的项数．注1 对每一个«Skip Record If...»，有«Skip Record If...»．注2 对任意两个正整数«Skip Record If...»，如果«Skip Record If...»，则«Skip Record If...»．反之，若«Skip Record If...»，则«Skip Record If...»．注3 «Skip Record If...»，有«Skip Record If...».注4 «Skip Record If...»«Skip Record If...»的任一子列«Skip Record If...»收敛于«Skip Record If...».3.数列有界对数列«Skip Record If...»，若«Skip Record If...»，使得对«Skip Record If...»，有«Skip Record If...»，则称数列«Skip Record If...»为有界数列．4.无穷大量对数列«Skip Record If...»，如果«Skip Record If...»，«Skip Record If...»，有«Skip Record If...»，则称«Skip Record If...»为无穷大量，记作«Skip Record If...»．注1 «Skip Record If...»只是一个记号，不是确切的数．当«Skip Record If...»为无穷大量时，数列«Skip Record If...»是发散的，即«Skip Record If...»不存在．注2 若«Skip Record If...»，则«Skip Record If...»无界，反之不真．注3 设«Skip Record If...»与«Skip Record If...»为同号无穷大量，则«Skip Record If...»为无穷大量．注4 设«Skip Record If...»为无穷大量，«Skip Record If...»有界，则«Skip Record If...»为无穷大量．注5 设«Skip Record If...»为无穷大量，对数列«Skip Record If...»，若«Skip Record If...»，«Skip Record If...»使得对«Skip Record If...»，有«Skip Record If...»，则«Skip Record If...»为无穷大量．特别的，若«Skip Record If...»，则«Skip Record If...»为无穷大量．5.无穷小量若«Skip Record If...»，则称«Skip Record If...»为无穷小量．注1 若«Skip Record If...»，«Skip Record If...»有界，则«Skip Record If...»．注2 若«Skip Record If...»，则«Skip Record If...»；若«Skip Record If...»，且«Skip Record If...»使得对«Skip Record If...»，«Skip Record If...»，则«Skip Record If...»．6.收敛数列的性质（1）若«Skip Record If...»收敛，则«Skip Record If...»必有界，反之不真．（2）若«Skip Record If...»收敛，则极限必唯一．（3）若«Skip Record If...»，«Skip Record If...»，且«Skip Record If...»，则«Skip Record If...»，使得当«Skip Record If...»时，有«Skip Record If...»．注这条性质称为“保号性”，在理论分析论证中应用极普遍．（4）若«Skip Record If...»，«Skip Record If...»，且«Skip Record If...»，使得当«Skip Record If...»时，有«Skip Record If...»，则«Skip Record If...»．注这条性质在一些参考书中称为“保不等号（式）性”．（5）若数列«Skip Record If...»、«Skip Record If...»皆收敛，则它们和、差、积、商所构成的数列«Skip Record If...»，«Skip Record If...»，«Skip Record If...»，«Skip Record If...»（«Skip Record If...»）也收敛，且有«Skip Record If...»«Skip Record If...»«Skip Record If...»，«Skip Record If...»«Skip Record If...»«Skip Record If...»，«Skip Record If...»«Skip Record If...»（«Skip Record If...»）．7.迫敛性（夹逼定理）若«Skip Record If...»，使得当«Skip Record If...»时，有«Skip Record If...»，且«Skip Record If...»«Skip Record If...»，则«Skip Record If...»．8. 单调有界定理单调递增有上界数列«Skip Record If...»必收敛，单调递减有下界数列«Skip Record If...»必收敛．9. Cauchy收敛准则数列«Skip Record If...»收敛的充要条件是：«Skip Record If...»，有«Skip Record If...»．注Cauchy收敛准则是判断数列敛散性的重要理论依据．尽管没有提供计算极限的方法，但它的长处也在于此――在论证极限问题时不需要事先知道极限值．10.Bolzano Weierstrass定理有界数列必有收敛子列．11.«Skip Record If...»12.几个重要不等式(1) «Skip Record If...» «Skip Record If...» «Skip Record If...»(2)算术－几何－调和平均不等式：对«Skip Record If...»记«Skip Record If...» (算术平均值)«Skip Record If...» (几何平均值)«Skip Record If...» (调和平均值)有均值不等式: «Skip Record If...»等号当且仅当«Skip Record If...»时成立. （3） Bernoulli 不等式: (在中学已用数学归纳法证明过)对«Skip Record If...»由二项展开式«Skip Record If...»«Skip Record If...»（４）Cauchy－Schwarz 不等式:«Skip Record If...»(«Skip Record If...»),有«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»（５）«Skip Record If...»，«Skip Record If...»13.O. Stolz公式二、典型例题1．用“«Skip Record If...»”“«Skip Record If...»”证明数列的极限．（必须掌握）例１用定义证明下列各式：（１）«Skip Record If...»；（２）设«Skip Record If...»，«Skip Record If...»，则«Skip Record If...»；（97，北大，10分）（３）«Skip Record If...»«Skip Record If...»证明：（１）«Skip Record If...»，欲使不等式«Skip Record If...»成立，只须«Skip Record If...»，于是，«Skip Record If...»，取«Skip Record If...»，当«Skip Record If...»时，有«Skip Record If...»即 «Skip Record If...»．（２）由«Skip Record If...»，«Skip Record If...»，知«Skip Record If...»，有«Skip Record If...»，则«Skip Record If...»«Skip Record If...»于是，«Skip Record If...»，有«Skip Record If...»«Skip Record If...»，即«Skip Record If...»．（３）已知«Skip Record If...»，因为«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»，所以，«Skip Record If...»，欲使不等式«Skip Record If...»«Skip Record If...»«Skip Record If...»成立，只须«Skip Record If...»．于是，«Skip Record If...»，取«Skip Record If...»«Skip Record If...»，当«Skip Record If...»时，有«Skip Record If...»«Skip Record If...»«Skip Record If...»，即«Skip Record If...»．评注１本例中，我们均将«Skip Record If...»做了适当的变形，使得«Skip Record If...»，从而从解不等式«Skip Record If...»中求出定义中的«Skip Record If...»．将«Skip Record If...»放大时要注意两点：①«Skip Record If...»应满足当«Skip Record If...»时，«Skip Record If...»．这是因为要使«Skip Record If...»，«Skip Record If...»必须能够任意小；②不等式«Skip Record If...»容易求解．评注２用定义证明«Skip Record If...»«Skip Record If...»，对«Skip Record If...»，只要找到一个自然数«Skip Record If...»，使得当«Skip Record If...»时，有«Skip Record If...»即可．关键证明«Skip Record If...»的存在性．评注３在第二小题中，用到了数列极限定义的等价命题，即：（１）«Skip Record If...»，有«Skip Record If...»（«Skip Record If...»为任一正常数）.（２）«Skip Record If...»，有«Skip Record If...»«Skip Record If...».例２用定义证明下列各式：（１）«Skip Record If...»；（92，南开，10分）（２）«Skip Record If...»«Skip Record If...»证明：（１）（方法一）由于«Skip Record If...»（«Skip Record If...»），可令«Skip Record If...»（«Skip Record If...»），则«Skip Record If...»«Skip Record If...»（«Skip Record If...»）当«Skip Record If...»时，«Skip Record If...»，有«Skip Record If...»«Skip Record If...»«Skip Record If...»即«Skip Record If...»．«Skip Record If...»，欲使不等式«Skip Record If...»«Skip Record If...»成立，只须«Skip Record If...»．于是，«Skip Record If...»，取«Skip Record If...»，当«Skip Record If...»时，有«Skip Record If...»«Skip Record If...»，即«Skip Record If...»．（方法二）因为«Skip Record If...»，所以«Skip Record If...»«Skip Record If...»，«Skip Record If...»，欲使不等式«Skip Record If...»«Skip Record If...»成立，只须«Skip Record If...»．于是，«Skip Record If...»，取«Skip Record If...»，当«Skip Record If...»时，有«Skip Record If...»«Skip Record If...»，即«Skip Record If...»．（２）当«Skip Record If...»时，由于«Skip Record If...»，可记«Skip Record If...»（«Skip Record If...»），则«Skip Record If...»«Skip Record If...»（«Skip Record If...»）当«Skip Record If...»时，«Skip Record If...»，于是有«Skip Record If...»«Skip Record If...»．«Skip Record If...»，欲使不等式«Skip Record If...» «Skip Record If...»«Skip Record If...»成立，只须«Skip Record If...»．对«Skip Record If...»，取«Skip Record If...»，当«Skip Record If...»时，有 «Skip Record If...» «Skip Record If...»«Skip Record If...»．当«Skip Record If...»时，«Skip Record If...»（«Skip Record If...»），而«Skip Record If...»«Skip Record If...»．则由以上证明知«Skip Record If...»，有«Skip Record If...»，即«Skip Record If...»，故 «Skip Record If...»．评注１在本例中，«Skip Record If...»，要从不等式«Skip Record If...»中解得«Skip Record If...»非常困难．根据«Skip Record If...»的特征，利用二项式定理展开较容易．要注意，在这两个小题中，一个«Skip Record If...»是变量，一个«Skip Record If...»是定值．评注２从第一小题的方法二可看出算术－几何平均不等式的妙处．评注３第二小题的证明用了从特殊到一般的证法．例用定义证明：«Skip Record If...»（«Skip Record If...»）（山东大学）证明：当«Skip Record If...»时，结论显然成立．当«Skip Record If...»时，欲使«Skip Record If...»成立，只须«Skip Record If...»«Skip Record If...»．于是«Skip Record If...»，取«Skip Record If...»«Skip Record If...»，当«Skip Record If...»时，有«Skip Record If...»即«Skip Record If...»．例设«Skip Record If...»，用“«Skip Record If...»”语言，证明：«Skip Record If...»．证明：当«Skip Record If...»时，结论恒成立．当«Skip Record If...»时，«Skip Record If...»，欲使«Skip Record If...»«Skip Record If...»只须«Skip Record If...»«Skip Record If...»．于是«Skip Record If...»，取«Skip Record If...»«Skip Record If...»，当«Skip Record If...»时，有«Skip Record If...»«Skip Record If...»即«Skip Record If...»．2.迫敛性（夹逼定理）«Skip Record If...»项和问题可用夹逼定理、定积分、级数来做，通项有递增或递减趋势时考虑夹逼定理．«Skip Record If...»，«Skip Record If...»，«Skip Record If...»«Skip Record If...»有界，但不能说明«Skip Record If...»有极限．使用夹逼定理时，要求«Skip Record If...»趋于同一个数．例求证：«Skip Record If...»（«Skip Record If...»为常数）．分析：«Skip Record If...»，因«Skip Record If...»为固定常数，必存在正整数«Skip Record If...»，使«Skip Record If...»，因此，自«Skip Record If...»开始，«Skip Record If...»，«Skip Record If...»，«Skip Record If...»，且«Skip Record If...»时，«Skip Record If...»．证明：对于固定的«Skip Record If...»，必存在正整数«Skip Record If...»，使«Skip Record If...»，当«Skip Record If...»时，有«Skip Record If...»«Skip Record If...»，由于«Skip Record If...»«Skip Record If...»，由夹逼定理得«Skip Record If...»，即 «Skip Record If...»．评注当极限不易直接求出时，可将求极限的变量作适当的放大或缩小，使放大、缩小所得的新变量易于求极限，且二者极限值相同，直接由夹逼定理得出结果．例若«Skip Record If...»是正数数列，且«Skip Record If...»，则«Skip Record If...»．证明：由«Skip Record If...»«Skip Record If...»，知«Skip Record If...»«Skip Record If...»即«Skip Record If...»«Skip Record If...»．于是，«Skip Record If...»«Skip Record If...»，而由已知«Skip Record If...»及«Skip Record If...»«Skip Record If...»故 «Skip Record If...»«Skip Record If...»由夹逼定理得 «Skip Record If...»．评注1 极限四则运算性质普遍被应用，值得注意的是这些性质成立的条件，即参加运算各变量的极限存在，且在商的运算中，分母极限不为0．评注2 对一些基本结果能够熟练和灵活应用．例如：（1）«Skip Record If...»（«Skip Record If...»）（2）«Skip Record If...»（«Skip Record If...»）（3）«Skip Record If...»（«Skip Record If...»）（4）«Skip Record If...»（5）«Skip Record If...»（«Skip Record If...»）（6）«Skip Record If...»«Skip Record If...»例证明：若«Skip Record If...»（«Skip Record If...»有限或«Skip Record If...»），则«Skip Record If...»（«Skip Record If...»有限或«Skip Record If...»）．证明：（１）设«Skip Record If...»为有限，因为«Skip Record If...»，则«Skip Record If...»，有«Skip Record If...».于是«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»．其中«Skip Record If...»为非负数．因为«Skip Record If...»，故对上述的«Skip Record If...»，有«Skip Record If...»．取«Skip Record If...»当«Skip Record If...»时，有«Skip Record If...»即«Skip Record If...»．（２）设«Skip Record If...»，因为«Skip Record If...»，则«Skip Record If...»，有«Skip Record If...»，且«Skip Record If...»．于是«Skip Record If...» «Skip Record If...» «Skip Record If...»«Skip Record If...»取«Skip Record If...»，当«Skip Record If...»时，«Skip Record If...»，于是«Skip Record If...»．即«Skip Record If...»（３）«Skip Record If...»时证法与（２）类似．评注１这一结论也称Cauchy第一定理，是一个有用的结果，应用它可计算一些极限，例如：（１）«Skip Record If...»（已知«Skip Record If...»）；（２）«Skip Record If...»（已知«Skip Record If...»）．评注２此结论是充分的，而非必要的，但若条件加强为“«Skip Record If...»为单调数列”，则由«Skip Record If...»可推出«Skip Record If...»．评注３证明一个变量能够任意小，将它放大后，分成有限项，然后证明它的每一项都能任意小，这种“拆分方法”是证明某些极限问题的一个常用方法，例如：若«Skip Record If...»，«Skip Record If...»（«Skip Record If...»为有限数），证明：«Skip Record If...»．分析：令«Skip Record If...»，则«Skip Record If...»．只须证«Skip Record If...»（«Skip Record If...»）由于«Skip Record If...»，故«Skip Record If...»，有«Skip Record If...»．于是«Skip Record If...»«Skip Record If...»再利用«Skip Record If...»（«Skip Record If...»）即得．例求下列各式的极限：（１）«Skip Record If...»（２）«Skip Record If...»（３）«Skip Record If...»解：（１）«Skip Record If...»«Skip Record If...»∵«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»，«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»，由夹逼定理，∴«Skip Record If...»（２）«Skip Record If...»∵«Skip Record If...»，由夹逼定理，∴«Skip Record If...»．（３）∵«Skip Record If...»，∴«Skip Record If...»．∵«Skip Record If...»«Skip Record If...»，由夹逼定理，∴«Skip Record If...»．评注«Skip Record If...»的极限是１，用此法体现了“１”的好处，可以放前，也可放后．若极限不是１，则不能用此法，例如：«Skip Record If...»，求«Skip Record If...»．解：∵«Skip Record If...»，«Skip Record If...»单调递减，«Skip Record If...»单调递减有下界，故其极限存在．令«Skip Record If...»，∵«Skip Record If...»∴«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»，«Skip Record If...»，∴«Skip Record If...»，即«Skip Record If...»．«Skip Record If...»（中科院）评注拆项：分母是两项的积，«Skip Record If...»插项：分子、分母相差一个常数时总可以插项．«Skip Record If...»３单调有界必有极限常用方法：①«Skip Record If...»；②«Skip Record If...»；③归纳法；④导数法．«Skip Record If...»«Skip Record If...»«Skip Record If...»单调递增«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»单调递减«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»不解决决问题．命题：«Skip Record If...»，若«Skip Record If...»单调递增，且«Skip Record If...»（«Skip Record If...»），则«Skip Record If...»单调递增（单调递减）．例求下列数列极限：（1）设«Skip Record If...»，«Skip Record If...»，«Skip Record If...»；（98，华中科大，10分）（2）设«Skip Record If...»，«Skip Record If...»；（04，武大）（3）设«Skip Record If...»，«Skip Record If...»，«Skip Record If...»（«Skip Record If...»）．（2000,浙大）解：（1）首先注意«Skip Record If...»，所以«Skip Record If...»为有下界数列．另一方面，因为«Skip Record If...»．（或«Skip Record If...»）故«Skip Record If...»为单调递减数列．因而«Skip Record If...»存在，且记为«Skip Record If...»．由极限的四则运算，在«Skip Record If...»两端同时取极限«Skip Record If...»，得«Skip Record If...»．并注意到«Skip Record If...»，解得«Skip Record If...»．（2）注意到«Skip Record If...»，于是«Skip Record If...»为有界数列．另一方面，由«Skip Record If...»«Skip Record If...»知«Skip Record If...»«Skip Record If...»．即«Skip Record If...»与«Skip Record If...»保持同号，因此«Skip Record If...»为单调数列，所以«Skip Record If...»存在（记为«Skip Record If...»）．由极限的四则运算，在«Skip Record If...»两端同时取极限«Skip Record If...»，得«Skip Record If...»．并注意到«Skip Record If...»，解得«Skip Record If...»．（3）由于«Skip Record If...»,又«Skip Record If...»«Skip Record If...»,所以 «Skip Record If...»«Skip Record If...»．评注１求递归数列的极限，主要利用单调有界必有极限的原理，用归纳法或已知的一些基本结果说明数列的单调、有界性．在说明递归数列单调性时，可用函数的单调性．下面给出一个重要的结论：设«Skip Record If...»（«Skip Record If...»）«Skip Record If...»，若«Skip Record If...»在区间«Skip Record If...»上单调递增，且«Skip Record If...»（或«Skip Record If...»），则数列«Skip Record If...»单调递增（或单调递减）．评注2第三小题的方法较为典型，根据所给的«Skip Record If...»之间的关系，得到«Skip Record If...»与«Skip Record If...»的等式，再利用错位相减的思想，将数列通项«Skip Record If...»写成级数的表达式．例设«Skip Record If...»为任意正数，且«Skip Record If...»，设«Skip Record If...»，«Skip Record If...»（«Skip Record If...»），则«Skip Record If...»，«Skip Record If...»收敛，且极限相同．证明：由«Skip Record If...»«Skip Record If...»«Skip Record If...»，知«Skip Record If...»«Skip Record If...»．则«Skip Record If...»，即«Skip Record If...»为单调有界数列．又«Skip Record If...»，且«Skip Record If...»«Skip Record If...»«Skip Record If...»，所以«Skip Record If...»亦为单调有界数列．由单调有界必有极限定理，«Skip Record If...»与«Skip Record If...»存在，且分别记为«Skip Record If...»与«Skip Record If...»．在«Skip Record If...»与«Skip Record If...»两端同时取极限«Skip Record If...»，得«Skip Record If...»与«Skip Record If...»．考虑到«Skip Record If...»为任意正数且«Skip Record If...»．即得«Skip Record If...»．例（1）设«Skip Record If...»，«Skip Record If...»，求«Skip Record If...»；（2）设«Skip Record If...»，«Skip Record If...»，且«Skip Record If...»（«Skip Record If...»），求«Skip Record If...»．解：（1）假设«Skip Record If...»存在且等于«Skip Record If...»，由极限的四则运算，在«Skip Record If...»两端同时取极限«Skip Record If...»，得«Skip Record If...»，即«Skip Record If...»．又«Skip Record If...»，故«Skip Record If...»．下面只须验证数列«Skip Record If...»趋于零（«Skip Record If...»）．由于«Skip Record If...»«Skip Record If...»，而«Skip Record If...»«Skip Record If...»，由夹逼定理得«Skip Record If...»«Skip Record If...»．（2）由«Skip Record If...»，知«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»，则«Skip Record If...»．假设«Skip Record If...»存在且等于«Skip Record If...»，由极限的四则运算，得«Skip Record If...»．下面只须验证数列«Skip Record If...»趋于零（«Skip Record If...»）．由于«Skip Record If...»«Skip Record If...»«Skip Record If...»．显然«Skip Record If...»«Skip Record If...»，由夹逼定理得«Skip Record If...»．评注１两例题中均采用了“先求出结果后验证”的方法，当我们不能直接用单调有界必有极限定理时，可以先假设«Skip Record If...»，由递归方程求出«Skip Record If...»，然后设法证明数列«Skip Record If...»趋于零．评注２对数列«Skip Record If...»，若满足«Skip Record If...»（«Skip Record If...»），其中«Skip Record If...»，则必有«Skip Record If...»．这一结论在验证极限存在或求解递归数列的极限时非常有用．评注３本例的第二小题还可用Cauchy收敛原理验证它们极限的存在性．设«Skip Record If...»>0，«Skip Record If...»＝«Skip Record If...»＋«Skip Record If...»，证明«Skip Record If...»＝1（04，上海交大）证（1）要证«Skip Record If...»＝1 ，只要证«Skip Record If...»，即只要证«Skip Record If...»，即证«Skip Record If...»（2）因«Skip Record If...»＝«Skip Record If...»＋«Skip Record If...»，故«Skip Record If...»，«Skip Record If...»«Skip Record If...»因此只要证«Skip Record If...»，即只要证«Skip Record If...»（3）由«Skip Record If...»知，«Skip Record If...»单调增加，假如«Skip Record If...»有上界，则«Skip Record If...»必有极限«Skip Record If...»，由«Skip Record If...»＝«Skip Record If...»＋«SkipRecord If...»知，«Skip Record If...»＝«Skip Record If...»＋«Skip Record If...»，因此«Skip Record If...»，矛盾.这表明«Skip Record If...»单调增加、没有上界，因此«Skip Record If...». （证完）４利用序列的Cauchy收敛准则例（1）设«Skip Record If...»（«Skip Record If...»），«Skip Record If...»，求«Skip Record If...»；（2）设«Skip Record If...»，«Skip Record If...»，«Skip Record If...»，求«Skip Record If...»；解：（1）由«Skip Record If...»（«Skip Record If...»），得«Skip Record If...»．假设«Skip Record If...»，则«Skip Record If...»．有«Skip Record If...»«Skip Record If...»由归纳法可得«Skip Record If...»．于是«Skip Record If...»«Skip Record If...»«Skip Record If...»（«Skip Record If...»）．由Cauchy收敛准则知：«Skip Record If...»存在并记为«Skip Record If...»，由极限的四则运算，在«Skip Record If...»两端同时取极限«Skip Record If...»，得«Skip Record If...»．注意到«Skip Record If...»，故«Skip Record If...»．（2）设«Skip Record If...»，显然«Skip Record If...».由于«Skip Record If...»,则«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...».于是«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»«Skip Record If...»(«Skip Record If...»).由Cauchy收敛准则知：«Skip Record If...»存在并记为«Skip Record If...».由极限的四则运算，在«Skip Record If...»两端同时取极限«Skip Record If...»，得«Skip Record If...»．注意到«Skip Record If...»，故«Skip Record If...»«Skip Record If...»．评注1 Cauchy收敛准则之所以重要就在于它不需要借助数列以外的任何数,只须根据数列各项之间的相互关系就能判断该数列的敛散性. 本例两小题都运用了Cauchy收敛准则,但细节上稍有不同.其实第一小题可用第二小题的方法,只是在第一小题中数列«Skip Record If...»有界,因此有«Skip Record If...».保证了定义中的N仅与«Skip Record If...»有关.评注2 “对«Skip Record If...»有«Skip Record If...»”这种说法与Cauchy收敛准则并不一致．这里要求对每个固定的«Skip Record If...»，可找到既与«Skip Record If...»又与«Skip Record If...»的关的Ｎ，当«Skip Record If...»，有«Skip Record If...»．而Cauchy收敛准则要求所找到的Ｎ只能与任意的«Skip Record If...»有关．５利用Stolz定理计算数列极限例求下列极限（1）«Skip Record If...»（2）假设«Skip Record If...»（00，大连理工，10）（04，上海交大）证明：Stolz公式«Skip Record If...»（3）«Skip Record If...»（4）«Skip Record If...»（5）«Skip Record If...»（«Skip Record If...»）６关于否定命题的证明（书上一些典型例题需背）«Skip Record If...»«Skip Record If...»发散例证明：«Skip Record If...»发散．例设«Skip Record If...»（«Skip Record If...»），且«Skip Record If...»，若存在极限«Skip Record If...»，则«Skip Record If...»．（北大，20）７杂例(1) «Skip Record If...»(2)（04，武大）«Skip Record If...»(3) «Skip Record If...»(«Skip Record If...»);(4)设«Skip Record If...»,«Skip Record If...»（«Skip Record If...»），求：«Skip Record If...»．。