数学竞赛之无穷级数

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ᦪᦪ4.1.ᭆᑁ⌕ᦪancanᦈឋ!஺#ᦪanbn%ᦈ&ᑣᦪ(anbn)*ᦈ&nInInInIn1,(anbn)anbrio
ᦪanbn%3ᦣ&ᑣᦪ(anbn)4563ᦣ஺nInInIn1nIn1#ᦪanᦈ&bn3ᦣ&ᑣᦪ(anbn)73ᦣ஺nInIn1ᵫᦪ(anbn)ᦈ49:ᑮᦪanbnᦈ஺nInIn1<=ᦪqn1,@q1Aᦈ,qn1nIn11B@q1A3ᦣ஺1qUPᦪp,@p>lAᦈ&@0p13ᦣ஺ᐸJKLᦪ3ᦣ஺nInnIn1ᦪ3ᦣ&ᐸJkNOPᦪ஺ᦪ(anan1)ᦈlimanRᙠ஺nnInInk᝞Uᦪanᦈ&ᑣliman0஺᝞Uliman0,ᑣᦪan73ᦣ஺nInnn1ᦋW5XᦪḄZ[ᨵ▲⚗&4ᦋWᐸᦣឋ&_ᙠᦈA`ᦪḄLᦋW஺ᦈᦪabcdeᦈf`ᦪL஺#abcdᡠ:ᦪ3ᦣ&ᑣ`ᦪ*3ᦣ஺

1O⚗ᦪhi:1.O⚗ᦪḄᦈkᑣlanᦈSnMn12.O⚗ᦪ=mᑨoilᜧᦈq7ᦈ&qᦣᜧ7ᦣ஺an#lim110&ᑣbnᦈanᦈBan3ᦣbn3ᦣ஺nbnInInInInanan#lim0,ᑣbnᦈanᦈ஺#lim&ᑣbn3ᦣan3ᦣ஺nbnbnInInInInn1r⚪APtᦪanpᦪpanḄᦣឋ஺nInInn13.᪷vᑨoilwl01A&ᦪᦈB@1A&nᦪ3ᦣB@1A&4x6஺y[l=0Aᦪ*ᦈ஺a4.=vᑨoilwllimn1,ᑣ@01A&ᦪᦈB@1A&nanᦪ3ᦣB@1A&4x6஺y[l=0Aᦪ*ᦈ஺5.zᑖᑨoilfx|ᙠ1,}ᓫK⌴ḄO⚗ᦪ&ᑣO⚗ᦪfnzᑖn11fxdxᐹᨵ!Ḅᦈឋ஺zᑖIfxdxḄᦣឋḄᑨoiO⚗ᦪḄ!஺n6.6ilsnulu2un;limsnRᙠ&ᑣᦈBᔲᑣ3ᦣ஺┯ᦪulu2u3u4(ᡈulu2u3,un0)Ḅhi---ᐻ6ᳮlunun1᝞U┯ᦪ&ᦪᦈ,ᐸLsul,ᐸ⚗rnḄvrnun1஺limuOnn

2┯ᦪ1n1nanᑨᦈ5ᵨiln(1)ᐻ6ᳮl᝞U┯ᦪanan1,liman஺ᦪᦈ,ᐸLsal,ᐸ⚗rnḄvrnan1஺᝞Uan4ᩩ&ᑣ5ᦋᵨl(2)¡¢⚗Ḅvᡠ᪀ᡂḄᦪ&#ᦈᑣ`ᦪᦈB#¥vᡠ᪀ᡂḄᦪᵨ=viᡈ᪷viᑨ63ᦣ&ᑣ¢⚗4¦f0,`ᦪ3ᦣ஺(3)§⚗ᡈ¨⚗Ḅi&t¢⚗§ᡂ©⚗&#ª¥©⚗ᑖo«¢⚗Ḅᦪᙳᦈ&ᑣ`ᦪᦈB#5ᦪᦈ53ᦣ&ᑣ`ᦪ3ᦣ஺#¨⚗dḄᦪ3ᦣ&ᑣ`ᦪ*3ᦣ஺(4)᝞U9®ᓽ°±liman0,ᑣᦪan73ᦣ஺nn1ᦈᩩᦈl#ananᦈ,²NᦈB#ananᦈᑣ²Nᩩᦈ஺n1nInIn1ᵫanan*3ᦣ஺_᝞UannInIn1³±Ḅ&ᑣliman0,´µliman0,f|an*3ᦣ஺nnn11(1)1KLᦪ3ᦣ&µ2ᦈ஺nnnᦈᦪḄLeᦈ&ᦈᦪᩩᦈᦪḄL|ᩩᦈ஺Z[⚗ᦪḄᑨoilজvᑨol#ᦪnaaᦈ஺ᓽᦈḄnnnIn1ᦪ56ᦈ஺ঝ§⚗ᡈ¨⚗Ḅi&t¢⚗§ᡂ©¸⚗L&ᑭᵨ┯ᦪLO⚗ᦪḄᑨoi஺ᐸ5ᑨoº»᝞¼ᡠ½lx1A&ᦈfx1A&3ᦣ11Xᦪ:

31xx2x3xnfᦪ(3)a0alxa2x2anxn,᝞UÊ4|¾ᙠ`ᦈ&*4|ᙠᐰxRAᦈᦪÀ}%ᦈ&ᑣ7RᙠR,ÁxRA3ᦣ&ᐸJR²NᦈÂÃ஺xRA4610A&RÄᦈÂÃḄilwlimnan1,ᐸJan,an1|(3)ḄÅᦪ&ᑣan0A&RÆA&R0ᦪᙠᦈÇ}ḄឋÈl#Éᦪaxnn1n1nḄᦈÂÃNRI,nbx

4nn1nḄᦈÂÃNR2,ᑣ(an1nbn)xanxbnxn,ᦈÂÃRminRI,R2஺

5nn11ËlÌᦪnxnḄᦈÇN2n22nlnl1,limn1,rlᵫffi\/n(஻+1)+1limḄᦈÂÃN1,nxnḄᦈnn2n2n2n22

61nlÂÃN2n-Jn,nxḄᦈÂÃN1,@xn-Jn1A&ᦪnxnᦈ&22n2n2ᡠª&ᦈÇN1,1஺@©XÍᦪḄᦈÇ4!A&ÊÎḄLḄᦈÇ|©XᦈÇḄÏ&ÐÑiªÒᓄÄÉᦪḄᦈÇ஺ÆᦪᙠᦈÇR,R}ᦈ&,LᦪS(x)Nᦪ஺#anxnᙠ-RᡈRᜐn1ᦈ&ᑣS(x)ᙠ-RᡈRᜐᑖoÕஹ×஺LᦪS(x)NØᦪ,Sxannxn1,⌲⚗ÄØdᦈÂÃ4W஺LᦪS(x)Nzᦪ,ஹn1Stdtn1antndt,⌲⚗zᑖdᦈÂÃ4W஺⌲⚗ÄØஹ⌲⚗zᑖd&ᦈÂÃ

74W_ᦈÇ9ᦋW&ᙠÚᜐḄᦣឋ9ᦋW஺#Éᦪaxnn1nnᙠXxOᜐ3ᦣ&ᑣ@xX3ᦣ஺᝞UᙠÞ0Aᦪanxn1xxOᜐßᦪᩩᦈ&ᑣxxO7àfáᗩᦪḄᦈÇḄÚ஺ann1xᙠx=3ᜐ(C)nInIn1A.ᩩᦈB.ᦈC.3ᦣD.ᦈឋanᐵËlwêᦪanx-1ᙠx=3ᜐᩩᦈ&ᑣÆᦪnrl`Íᦪᙠx=3ᜐᩩᦈëìᦈÂÃN37=2஺ᦪï⌲⚗zᑖஹðñd&ᦈÂÃ4W&ᡠªd5òᦪḄᦈÇN(-2,2KX=3ᙠᦈÇ᜜&ᡠªᙠáᜐ3ᦣ஺ÆᦪanxnᦈÂÃḄÄi:

8wlimn1nan1ᡈªN&ᑣ@nanB@0,AR஺¥ÑÄᦈÂÃḄi|ᐙᑖᩩ&0ARö@BöAR=01#limnan14RᙠA¨49ëᦈÂÃ4Rᙠ&÷NᦈÂÃø|RᙠḄ஺fùúan2nnanxஹanx3n<ᦪḄᦈÂÃ49Ð᪵ü&ý᪷þlimn1n1un11ÄᦈÂÃ஺un

92n!x2n2n!x2n,uḄᦈÂÃ஺rlwᵨ=vᑨoi&n22n1n!n!2n22n122n!x2nun1122xlimx4x4x1ᵫlim:l@A&ᦪᦈB22nun2n1n!n1n2n!x2nll2R@xA4x1,ᦪ3ᦣBᡠªᦈÂÃN஺222n1n!ËlÄnyfn┯rlᵫÿlimᦪn2n22n14,ᡠR1஺an1lim2n4ann1n3n2n1

10nx2nḄᦈᦪḄḄᡠᦪ⚗xnḄᦪan,"annḄ#ᓫᨵᳮ'ᑣ)*⚗+ᐸ*ᡂ.#ᓫḄᑖ01)⌲⚗3ᑖ'456ᑖ7ᑖ8Ḅn(ᡈnT,n+l>)1)⌲⚗1A'456ᑖ7ᑖBḄn(ᡈn-l,n+1>)1ᨬE4ᑭᵨᦪ0xn஺1xn0"anḄᑖBn!ᡈ2n!ᡈ2nT!JK)LMNᓄ#'ᨬEᑭᵨex,sinx,cosxḄUV஺"anḄᑖB2n!!ᡈ2n-l!!JK)LMNᓄ#'ᨬEᑭᵨ1xḄUV஺ᦪWXYᦈZ஺[ᵨN\]4sxanxn,ᵨ^ᑡ`an1mann1n1⌶dᦪsx(1)sx(nanx)dx1(2)sxXon1nIn1ᵨhᦪḄNijᦪ⚗ᦪḄk'⌕mᑮop〉rḄsᦪ'YtḄ'uvxiwxᑮyḄᦪ⚗ᦪḄ஺z{ij|oj}~ᐵḄk'ᐵk'+yḄn⚗ᑖ⊤ᡂz{ᑖ'ᯠE᩽▲஺x

11dᦪUVᡂᦪUVᑭᵨᦪ'+dᦪᙠipLUVᡂᢣḄᦪ஺f(xO)f(n)(xO)2dᦪUVᡂᦪf(x)f(xO)(xxO)(xxO)(xxO)n2!n!f(n1)()⚗Rn(xxO)n1,f(x)KUVᡂᦪḄᐙ⌕ᩩlimRn0n(n1)!f(0)2f(n)(0)nxO0kᓽ¡¢£᪍f(x)f(0)f(0)xxx2!n!fXUVᡂXḄᦪḄ1Yfnxn1,2,...;2fn0n1,2,...;f(0)2f(n)(0)nxx§YᦣR13ᑏYf(0)f(0)x2!n!f(n1)()n1limRnxlimx0«¬0}x0fxḄ4rxR,Rk'nn(n1)!f(0)2f(n)(0)n®¢£᪍ᦪᦈḄᐙ⌕ᩩ஺~kf(x)f(0)f(0)xxx2!n!UV)oḄ¯°(±⌕²³'ᦪ´¯°'⌲⚗3ᑖ⌲⚗A¯°)+dᦪµᓄᐸtdᦪ'¶·ᑭᵨ¸dᦪḄsᦪ(±⌕oj#ᓫdᦪḄ®¢£᪍UV)UV+¹ᩭdᦪUVᗩᦪ஺+dᦪUViᦪḄ±⌕N'ᐹ½N

12জᐜA'UVᡂiᦪEᙠ3ᑖ1ঝᐜ3ᑖ'UVᡂiᦪEᙠA஺rᯠ'7W⌕)oj〉rḄ¯°஺oj[ᵨdᦪUVᡂiᦪ:(1x)m1mxm(ml)2m(m1)(mnl)nxx1l)2!n!11x2n22n3!!xn1x12n!!n12n1!!n1xJl+ᦻ1x12n!!n11112n1!!xn1x1n12n!!In1xx21xn1x11x111xx2xn1x112xnxn11x121x1xx2xnIx2x3xnxIn1xxe1xX2n12!3!n!x3x5x2n1nsinxx1))3!5!(2nD!2nx2x4nxcosx11x2!4!2n!nIx2x3nxln1XX11x123n1eixeixeixeixixcosxsinxÄÅ:cosxisinx22

13ÆÇᦪ:aOf(t)A0Ansin(ntn)(ancosnxbnsinnx)2nIn1ᐸ7'aOaAO,anAnsinn,bnAncosntXoËÌឋ1,sinx,cosx,sin2x,cos2xsinnx,cosnxÏÐ`pÑÒ⚗Ḅ´3ᙠÓ]LḄ3ᑖ=0஺ᏝÕÖᦪ:aOf(x)(ancosnxbnsinnx),ᕜØ22n1ᐸ7an1f(x)cosnxdx,bn1(n1,2,3)f(x)sinnxdx112111212235822426224111211121222,12222346234122ËÛᦪ:an0,bnf(x)sinnxdxn1,2,3f(x)bnsinnx᜻dᦪ0Ûᦪbn0,an2Of(x)cosnxdxn0,1,2f(x)aOancosnxᏔdᦪ2aonxnxᕜØ21ḄᕜØdᦪḄᏝÕÖᦪf(x)(ancosbnsin),2n111InxlnxᕜØ21,ᐸ7anf(x)cosdx,bnf(x)sindx(n0,1,2,3)111111

14rXf(x)ḄÞßk'àᦪᦈ¬f(x)1rXf(x)Ḅák'àᦪᦈ¬f(X)ᙠàḄâã᩽▲Ḅäᙳw11fXfXo24.2.]⚪⌱è1Ḅ஺2nnIn1tanxtany,êᵫ¬tanxy1tanxtany]Líîᦪarctanrxy2kᨵxyarctantanxyarctantanxtany1tanxtany11111

15arctan2arctanarctanarctannnInn11nn11arctan2nn14n1]2.4Unᓫð⌴òóᨵḄËᦪᦪᑡ'ôõᦪôõᵫ¬Unᓫð⌴ò'ᑣ01nUnUn1Un1ᦈ஺Un1n1UUnUn1Unn1,Un1U1Ui1UiUn1UI,ᵫUnᓫð⌴òóᨵxlimUnöᙠnUlUli1Un1UnUnᦪᦈ'ᦪ1ᦈUlUn1nIn11

16Ḅ஺22nn1nn1Inn1ê2tanarctanarctan2nlnInnIn]3.ᦪarctanInn1arctanarctan2n2nInInarctan2limarctan2nnn14n1arctanann!]4.÷øᦪna0Ḅᦣឋ஺nInan1n1!ê:limnn1nlann!nnl1alim1alim1nnn1n1nnn1n1aera>ek'ᦪùᦣ1rk'ᦪᦈ஺1214161xxx…+x2n…ḄᦈZ§ᐸ.]5.ᦪ1234562n(2n1)112n஻2n2x,ᑣsxx,x1,1ê4sx21xn12n(2nl)n1dxl1xs'xIn,21x21xr11xl1xxllxl2sxInxlndxxlnInix21x21xlx221x21111X1k'SX2nnn12n(2nl)n12n1111...111Idxll1limlimln2ᦈOnnIn2nnnnlxZ1,1,dᦪ1xlIxlnIn1x2,x1,1sx21x2ln2,x1

17f(x)]6.4dᦪf(x)ᙠX0ḄiûZᑁᐹᨵý▤ÞAᦪ'ólim0,ôõᦪ1fÿᦈ஺n1nf(x)ᦪf(x)ᙠx0Ḅᑁᐹᨵ▤ᦪlim0,ᑣxxf(x)f(0)0oxf(x)f(x)f(x)f(0)1ᵫᑣlim2lim,limxOxOx0x2x221ff(0)1nfᡠlim,ᵫ!"ᑨ$%&ᦪ'ᦈ஺xOn12nn2f(0)limf(x)0,f(0)limxx2ᦪf(x)ᙠx0Ḅᑁᐹᨵ▤ᦪᑣf(x)ᙠ)ᑁḄ*+,-a,a
.ᨵ/ᓽ1ᙠ2ᦪM0,34f(x)M஺ᵫ5678f(x)2f(x)2xx,012!2Mx2%ᙠ,-9a,a
±,f(x),;<1ᙠ=᦮ᦪN,?nN@ឤᨵ211MlfoᵫB"ᑨ$%&ᦪf'ᦈ஺2n1nn2nanC7.DanOxsinxx,n1,2,EFnḄG஺nn12HIJxnt,ᑣOnnann(nt)sintdtnOsinxdxOxsinxdx,nnn22ᡠansinxdxsinxdx:n,n1,2,002212nnKS(x)nx,11,x1x1nIn11xin1⌲⚗F4nx1x1,2n1(1x)xn᦮ᳮ4nx1xlo2n1(1x)1x2n1OP⌲⚗F4nx1x13n1(1x)

18x(lx)2n᦮ᳮ4S(x)nx,1x1஺3n1(1x)f(x)f(0)f(0)xC8.DaO4,al1,an2n(nl)an,n2,(1)FQ&ᦪanxnḄRᦪS(x)*(2)n0FS(x)Ḅ᩽G஺HI(1)D&ᦪanxnḄᦈ,-(R,R),⌲⚗F4n0S(x)nanx,S(x)n(nl)anxn2,x(R,R)nInIn2Uo⚪W4S(x)an2xn2n2anxn,ᡠᨵS(x)S(x)0஺n0HB▤2XᦪYPZឋ\ᑖ^4S(x)CeC2e஺_ᐭabᩩdlxxS(0)aO4,S(0)al1஺535x3x4Cl,C2஺efS(x)ee஺22225x3x5x3xl3(2)JS(x)ee0,4xlngS(x)ee0,ᡠoS(x)ᙠ25222213xInᜐi᩽jG஺25C9.Df(x)f(0,).⌴mḄᦪf(x)0,nᦪᑡ/an0ᦈᐸqanf(k)f(x)dxk1InnoQan+l-anf(n1)f(x)dx(f(n1)f(x))dx0;nnnIn1(Qf(n1)f(x)0)gQan[f(1)f(x)dx][f(2)f(x)dx]K1223[f(n1)f(x)dx]f(n)0anᦈnInlnlllrandx1InnᦈsḄ᩽▲GKC,klklx2n

1911vwx2ᦪ1lnn=C+o1o1⊤j.2nnwx78IHnInncol,ᐸqHnimo1⊤jᓽInIlnInc஺wx2ᦪc,ᐸG0.57721566490153286060651209,%⍝sfᨵᳮᦪfᳮᦪ஺ᙠ\ᑖqwx2ᦪᨵᵨ᝞F£ᦪᑡḄ᩽▲஺ᵨwx78FHᨵᐵ&ᦪḄ¥⚪IIkIkn,c0.57721...,111...ḄR஺1F&ᦪnInIn21111Hn1HnHInInIn2nIn2n1HlnHn111Hl11nInnIn2n1nIn2222Fᑖ1110xxdxoHII1111201x1xdxlxldx1231x2dx...11nnlIn1n1xndxn1

20In1nn1INInn11NInN1H1nn1N11nIlimNINlimNInN1HN111c(3)ᑨ©&ᦪ1n1311...1Ḅᦣឋ2n1H9111112...1nlim1311131nnco12...n3co1

21Inln3,limnn3col13cnln3ln31,11n1nln3ᦈᦈn13112...InC᝞«¬᪵FᦪᑡxlnnIn112nḄ᩽▲G:

22ᵫ1121nlnn=C+o(1),R1121nlln12nln2n=C+o(1),¯8°m4Ixn-ln2o(1),xnln2.±2²«3ᵨ©ᑖḄ©³Flimnxn:limlllnxnlimnnlim(nn12n)limlnn(l1)1nln111xdxln20C10.Dᦪ£&)ᙠ().ᨵ©³ᙠx0Ḅ¶ᑁᨵ·▤ᦪf(x)lln,n(l)f()ᦈ

23nInInInnn462n2+ᦈ»FRC11.n3!5!(2n-1)2n22n1!2lim0,&ᦪᦈn:lim2nnn2n2n12n1!x2nx2n11xxexexDsx2n12nl!n12n1!se2eᓽᡠFR211nan1,afOC12.Dfx,F:&ᦪ஺n21xn!nOanan21nnf0n:Ãf(x)ᢥÅÆÇ᪍ÉÊ4fxanx,ᐸqann!n0fX11xx211xx2fx1x2anxnn011xx2aan0alxnxn2aOalaOxan2an1anxn2n0aOal1,an2an1an0an2an1an0an1anaO1,an2an1anan11,ann,limnan

24an1an2aInOan1nan2nOanan2n0anan+2=limn1+111aa=20aln+lan+2ᦑ&ᦪanla,nan2an1=2nOnOanan2C13.ᑨÌ&ᦪ+2I1ntann1Ḅᦣឋ஺>2+2ÎHIDun

25tanntan0,ulnn⌴mnimun0f+2Î1tann1fᦈḄ¿┯&ᦪ஺/?+yjn~+2஻++2

26n+\Jir+2/72Gu1ntanInn<&ᦪ1¸ᦣun¸ᦣᡠÓ&ᦪᩩdᦈ஺nInn1C14.nI1n12x2n1

27cosnx1nln2124,x,,»F&ᦪn1n2nIÃx2ᙠ.ÉÊᡂÕÖ&ᦪᑣa2200x2dx32,gan2On2x22xlxcosnxdxsinnx2cosnx3sinnxnnn0241,bn0,n0,1,2,...2nᡠx2ḄᏝØÙ&ᦪx2᦮ᳮ4234

28n11n2ncosnx,x,n1n11n2n11x2cosnx,x,Jx=04124n2n12212

29C15.Dᦪfxᙠx1.ᨵ©³ᙠx0ḄᑁᨵḄ▤ᦪx0@bn1fx0,?x0@fxfxḄÜ▤jnNᨵbnnI&ᦪ1fn1,ᵨ1fnn1x0@fxfxḄÜ▤jf00,fn:ᑁÃfxÉᡂ5678ᨵfxf0f00,ᙠx0Ḅfx2f஻x2,Þ2212஻஻e0,xß-,X,M,fxMx2

30MIMI?nᐙᑖᜧ@ᨵf2,2ᦈfᦈnn2nn12nn1111fffbbnlb2n133gbnblflb211ffn21fbn,bblf1äå4nnblflfln0xfᦈ,bnᦈInIflbnbn11ᦈ,2nIn12C16.4.3æç⚪x4n1.FQ&ᦪḄRᦪS(x)஺

31n0(4n)!n2ln2.Fê&ᦪnxḄRᦪS(x)஺n02n!3.DQ&ᦪa(x1)Ḅᦈ(4,2),ᑣë&ᦪna(x3)nnOnnOnnḄᦈ,-Ix2nx2n4.F&ᦪḄᦈ,-íJsx,FS(x)dxoOn0(2n)!n0(2n)!5.íbbnn1n0ᦈanan1ᦈnIanbn'ᦈ஺nIn16.Fïᑡ&ᦪḄᦈI(1)n11x13n12nnnÎ(2)1xn12n11x1nn7.Fð&ᦪn3n1In2nxnḄᦈ஺