数学分析数学分析数学分析 (11).pdf

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1、?1.?(?)?1?!21+31?!41+?51?;?11)1(nnxn?xn?11sin)1(nnnx;?11)1(nnnn;?22ln)1(nnnn;?13cos1nnn;?121sin4)1(nnnnnx;?1)1cos()1sin(npnxnxn;?nnnnxn?1212)1(;?()$%&?11)23)(23(12ln)1(nnnnn;?2lnnqpnnnx;?111)1(nnnaan().0?a?1?1?!21+31?!41+?51?nS?nknknkkS112)!2(1121?1)!2(1nn?1121nn?nnS2lim?1?!21+31?!41+?51?2?11)1(nnxn

2、?xn?n0?xn?xn1?11)1(nnxn?xn?xnn?1)1(?n1)(?n,?11nn?11)1(nnxn?xn?1?3?0?x?11sin)1(nnnx?0?x?11sin)1(nnnx?n?xn2?nxsin?11sin)1(nnnx?nxnsin)1(1?nx)(?n?1nnx?11sin)1(nnnx?4?1lim?nnn?nnnn1)1(lim?11)1(nnnn?5?22ln)1(nnnn?8.nnn2ln?22ln)1(nnnn?22lnnnn?22ln)1(nnnn?6?13cos1nnn?nS?nS6?nkkk211232)1(?nkkk21132)1(?nkkk2

3、13)1(?11232)1(nnn?1132)1(nnn?13)1(nnn?Leibniz?nnS6lim?16lim?nnS26lim?nnS36lim?nnS46lim?nnS56lim?nnSnnS6lim?13cos1nnn?nnn213cos1.?121nn?13cos1nnn?2?7?)6,6(?kkx?nnnnxnnx)sin4(1sin4)1(221?1sin402?x?12)sin4(1nnxn?121sin4)1(nnnnnx?6?=?kx?41sin2?x?121sin4)1(nnnnnx?11)1(nnn?nnnnxnnx)sin4(1sin4)1(221?1sin42

4、?x?8?2?kx?1)1cos()1sin(npnxnxn?2?kx?1?pppnnxnxn1)1cos()1sin(?1)1cos()1sin(npnxnxn?10?p?pnxnxn)1cos()1sin(ppnxnnx22sin22sin?Dirichlet?122sinnpnnx?122sinnpnx?1)1cos()1sin(npnxnxn?0?p?1)1cos()1sin(npnxnxn?3?9?nnnnxnx2)1(21?2limxxnnn?2?x?nnnnxn?1212)1(?2?x?nnnnxn?1212)1(?2?x?nnnnxn?1212)1(?10?1ln 2(32)(

5、32)nnunn&)?$%(?nu?11(1)nnnu?Leibniz?nun32ln)(?n?132lnnn?11(1)nnnu?11?nnxxqpnnln?xxnnn?lim?1?x?2lnnqpnnnx?1?x?2lnnqpnnnx?1?x?2lnnqpnnnx?2ln1nqpnn?1?p1,1?qp?1?x?2lnnqpnnnx?2ln)1(nqpnnn?1?p1,1?qp1,1?qp10?p?0,0?qp?12?nnnaanx?1)1(1?4?1?a11lim?axnnn?111)1(nnnaan?1?a?111)1(nnnaan?112)1(nnn?10?a?11)1(nnn?na

6、a1?Abel?111)1(nnnaan?nx?na)(?n?1nna?2.?Cauchy?1+21?31+41+51?61+71+81?91+?;?1?21+31+41?51+61+71?81+91+?1?nxnnnxxx62313?261431131?nnn?6126?nn?Cauchy?n?2?nxnnnxxx62313?nnn61631331?616?nn?Cauchy?n3.?Cauchy?=0?1nnxnxnnnx?lim?1nnx0?1?0?NNnm?20211?mnnxxx?)1(2?NNNn?2Nn?!#+,-?2201221?!#+,-!#+,-nnnnxxxxn?5?1?

7、nnx0?4.?0?1?p?),(pN1?+?+?1?nx2?nxpnx?1?n?N?1nnx?1nnx?1nnx?11nn?0?1?p?11ppN?),(?),(pNn1?1?121npxxxpnnn?5.?1nnxlimn?nnyx=1?1nny?1nny?nxnn1)1(?,nnynn1)1(1?limn?nnyx=1?1nnx?1nny6.?=0?0.nxlimn?nxnnnx?11)1(?11)1(nnnx?121212knkknkxn?0.nxlimn?0nx?nnnx?11)1(7.?nx?1)1(nnnx?()$%&?111nnnx?6?()$%&?111nnnx?Leibni

8、z?nx0lim?nnx?1)1(nnnx0lim?nnx?n?nnnx()$%&?()$%&?21111?()$%&?111nnnx?8.?10nnnx?0?1nnnx?1nnnx?)1(100?2?nnnnx?10nnnx?01?n?1nnnx?Dirichlet?9.?nnx?21)(nnnxxn?1nnx?,?Abel?,nnnaxb?1?kbBkiik?1?nknkkknkxxknxx1111)(?11)(nnnxxn1)(1(11?2?nnxxnnnn?1nn?Abel?11)(1(nnnxxn?21)(nnnxxn?11)(nnnxxnnnx?1nnx 710.?21)(nnnx

9、x?1nny?1nnnyx?,?1nny0,NNnNp1?/?0/?/?1?pnnkky1?,?,?21)(nnnxx?nx?Axxnnn?21,Bxn?,?knnnknyyyB?21,?Abel?1)()(1111BABxxBxyxpnnkkkkpnpnpnnkkk?Cauchy?1nnnyx11?)(xf 1,1?0)(lim0?xxfx?()$%&11nnf?0)(lim0?xxfx?0)0(?f?0)0(?f?()$%&nf1?212)0(nf2?n?()$%&11nnf?12?1nnx?()$%&?111nnxn?nnxny)11(?1nny?1nn?Abel?,?1nnx?11nn

10、ynn?()$%&?111nnxn?13.?0?nnxlimn?()$%&?11nnxx?0?nnnx?11)1(8?01lim1?()$%&?nnnxxn?n1?nnxx?nxn0,0?0?n?nnnnxxnn)1()11(11?1)1(?nxn?nxn?nxn?n?,?0?AAxnn?nAxn?0?Leibniz?nxnnnx?11)1(14.?1+21+31+?+n1?ln n?(?n)?Euler?(?2.4.8)?11)1(nnn?1+31?21+51+71?41+91+111?61+?.?nb112?+31+?+1lnnn?1+31?21+51+71?41+91+111?61+?n

11、nn21141341?nS?)ln(213nbSnn1+31+51+71+91+111+?141341?nn?3211(ln)(ln2)ln422nnnnSbnbnb?4n?92ln232121243?nnnnbbbS?nnblim?2ln23lim3?nnS?13limnnS?23limnnSnnS3lim?2ln23lim?nnS?15.?Cauchy?(1)2?0!1nn?0!)1(nnn=1?(2)=()$%&?0nnq()$%&?0nnq?0)1(nnqn2)1(1q?(?q?1?)?1?2?0!1nn?0!)1(nnn?0nnc?10?c?1.n?2?njijnjic!)1(jnjijinn)1(!1?2?0)11(!1?nn?2?0!1nn?0!)1(nnn=1?2?=?()$%&?0nnq()$%&?0nnq?0nncnnjijinqnqqc)1()(?1?q?qqnn?110?()$%&?0nnq()$%&?0nnq=?0)1(nnqn2)1(1q?)1(?q?10