数学分析数学分析数学分析 (15).pdf

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1、?10.3?1.?1)2(3nnnnxn;?nnxn)1(12111?()$%&?;?2?122)1(nnnnnx;?1)1(1)1ln()1(nnnxnn?nnnxn()$%&?21!31;?222lnnnnxnn?;?nnnxnn?1!?nnxnn?12!)2()!(;?nnxnn?1!)12(!)!2(?1?1)2(3nnnnxn?1nnnxa?3lim?nnna?31?R?31?x?1nnnxa?1)32(1 1nnn?31?x?1nnnxa?1)32()1(1nnnn?()+,-?31,31D?2?nnxn)1(12111?()$%&?1)1(nnnxa?1lim?nnna?1?R?

2、2?xnnnxa)1(1?()$%&?11211nn?0?xnnnxa)1(1?()$%&?11211)1(nnn?2,0?D?52?3?2?122)1(nnnnnx?1nnnxa?nnnalim212)1(lim2?2?nnnnn?2?R?2=?x?1nnnxa?1)1(nnn?*2,2?D?4?1)1(1)1ln()1(nnnxnn?1)1(nnnxa?1lim?nnna?1?R?0?x?1)1(nnnxa?11)1ln()1(nnnn?Leibniz?2?x?1)1(nnnxa?11)1ln(nnn?0,2?D?5?nnnxn()$%&?21!31?1)1(nnnxa?nnnaa1lim

3、032!2)!1(3lim11?!#+,-222?nnnnnnn?R,?,D?6?222lnnnnxnn?1nnnxa?nnnalim1lnlim22?nnnnn?1?R?1=?x?1nnnxa*1,1?D?7?nnnxnn?1!?1nnnxa?nnnaa1limennnnnnn1!)1()!1(lim1?!#+,-2?eR?ex=?1nnnxa?=?1)(!nnnenn?Stirling?!n?nnen?212?)(?n?53?nnenn)(!=?eeD,?8?nnxnn?12!)2()!(?1nnnxa?nnnaa1lim41)!()!2()!1(2)!1(lim22?!#+,-2?nnn

4、nn?4?R?4=?x?1nnnxannnn)4(!)2()!(12=?Stirling?!n?nnen?212?)(?n?nnn)4()!2()!(2=?4,4?D?9?nnxnn?1!)12(!)!2(?1nnnxa?nnnaa1lim1!)!2(!)!12(!)!32(!)!22(lim?!#+,-?2?nnnnn?1?R?1?x?1nnnxa?1!)12(!)!2()1(nnnn?Leibniz?1?x?1nnnxa?1!)12(!)!2(nnn?!)!12(!)!2(?nnbn?)1(lim1nnnbbn21?Raabe?*?1,1?D2.?a?b?0?nnnnxnbna?()$%&

5、?12;?1nnnnbax;?a x+b x2+a2 x3+b2x4+?+anx2n-1+bnx2n+?1?anbnannnn?()$%&?2lim?aR1?ax1?nnnnxnbna?()$%&?12?()$%&?12)1()1(nnnnnanbn?54?ax1?nnnnxnbna?()$%&?12?()$%&?121nnnanbn?()+,-?aaD1,1?2?abannnn11lim?aR?ax=?1nnnnbax?),(aaD?3?a x+b x2+a2 x3+b2x4+?+anx2n-1+bnx2n+?1nnnxc?nnnclimaannn?12lim?aR1?ax1=?1nnnxc

6、()$%&?aaD1,1?3.?R?0nnnxa?0nnnxb1?R2,?(1)?;(2)?;?02nnnxa?0)(nnnnxba(3)?0nnnnxba?1?02nnnxaR?1Rx?02nnnxa1Rx?02nnnxa1RR?2?0)(nnnnxbaR?55?x?21,minRR?0)(nnnnxba?x?21,minRR?21RR?0)(nnnnxba?1?21RR?0)(nnnnxba?0nnnxa?0nnx?0nnnxb?()$%&?0121nnnx?1?0)(nnnnxba2?21,minRRR.?3?0nnnnxbaR?nnnnbalimnnna?limnnnb?2lim?21

7、RRR.?1?0nnnxa?02nnx?0nnnxa?012nnx?1?0nnnnxba?R?4.?1nnnx?0212nnnx?121)1(nnnxn?1)1(nnnnx?1)1(nnxnn?12)!2(1nnnx?1!1nnxnn?1?1nnnx1?R?1=?x?)1,1(?D?1)(nnnxxS?11)()(nnnxxxSxf?56F?xdxxf0)(?F?110nnxdxnxxxxnn?11?()$%&?xxdxdxxS1)(2)1(xx?2?0212nnnx?1?R?1=?x?)1,1(?D?0212)(nnnxxS?)()(xxSxf?01212nnnx?20211)(xxxfnn

8、?F?xxdxxxS0211)(xxx?11ln21?3?121)1(nnnxn1?R?1=?x?)1,1(?D?121)1()(nnnxnxS?1121)1()()(nnnxnxxSxf?1?F?xdxxf0)(?F?11201)1(nnxndxxn?11)1(nnnnx2)1(xx?()$%&?2)1()(xxdxdxxS3)1()1(xxx?4?1)1(nnnnx?1?R?1=?x?57?1,1?D?1)1()(nnnnxxS?)()(xxSxf?11)1(nnnnx?xxxfnn?11)(11?)(xf)1ln(10 xxdxx?F?)(xSF?xdxxfx0)(1)1ln()11(1

9、xx?1,1)x?11(1)1(1)nSn n?1(1)S()S x,1?1(1)lim()1xSS x?5?1)1(nnxnn1?R?1=?x?)1,1(?D?1)1()(nnxnnxS?11)1()()(nnxnnxxSxf?1?F?xdxxf0)(?F?110)1(nnxdxxnn?1)1(nnxn1)1(12?x?()$%&?1)1(1)(2xdxdxxS3)1(2xx?6?12)!2(1nnnx?R?),(?D?12)!2(1)(nnnxxS?112)!12()(nnnxxS?xexSxS?)()(xexSxS?)()(?58)(21)(xxeexS?7?1!1nnxnn?R?),(

10、?D?1!1)(nnxnnxS?F?xdxxS0)()1(!11?xnnexnx?*?)1()(xexdxdxS1)1(?xex?10.3.6?1!1)(nnxnnxS?11)!1(nnnxx?1!1nnxn1)1(?xex?5.?f(x)=,?x=r?0nnnxa?0nnnxa?011nnnxna?x=r?Frxxf0d)(=?011nnnrna?F2?10d11lnxxx=?121nn?011nnnxna?x=r?011nnnxna?r?0nnnxar?*?rx,0?F?xnnnxnadxxf0011)(?101?nnnrna?,?101?nnnxna?*r,0?,?rx?F?rnnnrn

11、adxxf0011)(?xxxf?11ln1)(?59F?2?1011lnxdxxF?1011nndxnx?F?1101211nnnndxnx?6.?(1)y=?04)!4(nnnx?y(4)=y;(2)y=?02)!(nnnx?xy +y?y=0?1?4?)4(y?144)!44(nnnxynxnn?04)!4(?2?11!)!1(nnnnxy?22!)!2(nnnnxy?1yxy?21!)!1(nnnnnx?221)!1(1nnnxynxnn?02)!(?7.?112)1(nnnn;?2121nnn;?114)2(nnnn;?022)1(nnn;?0)12(31)1(nnnn;?22)1(

12、21)1(nnnn;?01!2)1(nnnn?1?11)1()(nnnnxxf?111)1()()(nnnnxxxfxg?2)1(1)(xxg?2)1()(xxxf?92)21(2)1(11?fnnnn?60?2?11)(nnxnxf?xxf?11ln)(?2121nnn?)21(f2ln?3?211)1(1xnxnn?11)2()(nnxnnxfF?xdxxf0)(2312)1(xxnxnn?32)1()3()(xxxxf?)41(4)2(11fnnnn2711?4?02)1()(nnxnxfF?xdxxf0)(?01)1(nnxn21)1(xxnxnn?3)1(1)(xxxf?022)1(

13、nnn12)21(?f?61?5?0212)1()(nnnxnxf?01212)1()()(nnnxnxxfxg?xxgarctan)(?xxxfarctan)(?63)31()12(31)1(0?fnnnn?6?)1ln()1(11xxnnnn?221)1()(nnnxnxf?2121)1()()(nnnxnxxfxg?)(xg?21)1(nnnxn)1ln()1(111xxxnnnn?F?xdxxxxxf0)1ln(1)(2141)1ln()1(21?xxxx?22)1(21)1(nnnn)21(f?23ln4383?7?01!)1()(nnnxnxf?()()f xg xx?0(1)()

14、!nnxng xxen?()()xf xxg xxe?01!2)1(nnnn?)2(f22e?628?1nna?nkknaA10lim?nnnAa?1nnnxa?nnnxa?11RnnnxA?12RnnAa?0?21RR.?1nna11?R?1limlim1111?nnnnnnnAaAAA?,?12?R11?R?9?122)(nnnxnxf?1?)(xf!#+,-?21,21?()+,-?21,21?2?)(xf21?x?1?122nnnxn?21?R?21=?x?Abel?122nnnxn?!#+,-?21,21?122)(nnnxnxf?!#+,-?21,21?()$%&nnxndxd2212?nnxn?0?3?112nnnxn?!#+,-?321,21?112nnnxn?()+,-?21,21?122)(nnnxnxf?()+,-?21,21?112)(nnnxnxf?2?)(xf21?x?,?xt2?122)(nnnxnxf?12nnnt?12)(nnntxg?63?F?tduuutg0)1ln()(?LHospital?1,1?t21)21()(lim21?xfxfx!#+,-?FF?ttduuuduuut0101)1ln()1ln(12lim?F?tttttduuut111)1ln(2lim)1ln(12lim?64