第十章（1）.pdf

?10.11.?:(1)?Pn=1cosnn(n+1)?.?.|n+pPk=n+1coskk(k+1)|n+pPk=n+11k(k+1)=n+pPk=n+1(1k1k+1)=1n+11n+p+1 0,?N=1,?n N?,|n+pPk=n+1coskk(k+1)|1n 0,?Pn=1an?Pn=1bn?,?N N,?n N?,|n+pPk=n+1ak|,|n+pPk=n+1bk|.?,n+pPk=n+1akn+pPk=n+1ukn+pPk=n+1bk.?|n+pPk=n+1uk|0,?limnS2n=limnS2n+1=A,?N 0,?n N?,|S2n A|,|S2n+1 A|N?,|Sn A|0,?Pn=1un?,?,?N,?n N?,n+pPk=n+1un N?,(2n)u2n 22nPk=n+1uk.(2n+1)u2n+12nPk=n+1uk+2n+1Pk=n+1uk 1,?,?.(4)Pn=11n1+1/n.?limn1n1+1/n/1n=1,?Pn=11n?,?.(5)Pn=1n2(31n)n.?limnnqn2(31n)n=13,?,?.(6)Pn=2n(lnn)n.?limnnqn(lnn)n=0,?,?.(7)Pn=11000nn!.?limn1000n+1(n+1)!/1000nn!=0,?,?.(8)Pn=1(n!)2(2n)!.?limn(n+1)!)2(2n+2)!/(n!)2(2n)!=14,?,?.(9)Pn=113n(n+1n)n2.?limnnq13n(n+1n)n2=e3 0).?p 1?,?R+21x(lnx)p=(lnx)1p1p|+2?,?.?p=1?,?R+21x(lnx)p=lnlnx|+2?,?.?0 p 0).?limn1n(lnlnn)q/1n(lnn)1/2=limn(lnn)1/2(lnlnn)q=limx+x1/2(lnx)q=+.?Pn=31n(lnn)1/2?,?.3.?:?Pn=1un?,?Pn=1u2n?.?,?.?.?Pn=1un?,?limnun=0,?un?.?un M.?u2n Mun.?,?Pn=1u2n?.?,?Pn=1(1n)2?,?Pn=11n?.4.?:?Pn=1a2n?Pn=1b2n?,?Pn=1|anbn|,Pn=1(an+bn)2,Pn=1|an|n?.?.?,?Pn=1(a2n+b2n)?.?|anbn|12(a2n+b2n),(an+bn)232(a2n+b2n),?Pn=1|anbn|,Pn=1(an+bn)2?.?bn=1n,?Pn=1|an|n?.5.?:?Pn=1un?Pn=1vn?,?(1)Pn=1(un+vn);(2)Pn=1(un vn);(3)Pn=1unvn.?.(1)?,?un+vn un,?,?Pn=1(un+vn)?.(2)?,?un=vn=1n?,?Pn=1(un vn)?0.(3)?,?un=vn=1n?,?Pn=1unvn?.6.?limnnun=l,?0 l 0?n?un 0.?un 0.?limnu2n/1n2=l2,?Pn=11n2?,?Pn=1u2n?.?limnun/1n=l,?Pn=11n?,?Pn=1un?.?10.31.?(1)Pn=1(1)n1(2n)2.?Pn=1|(1)n1(2n)2|=Pn=11(2n)2?,?.(2)Pn=1(1)n+1(2n1)p(p 0).?1(2n1)p?0,?.?Pn=11(2n1)p?p 1?,?p 1?,?.(3)Pn=2(1)nnlnn.?1n lnn?0,?.?Pn=21nlnn?,?.(4)Pn=1(1)nn1n.?=Pn=1(1)n 1nPn=1(1)n 1n,?,?.?limnn1n/1n=1,?Pn=1n1n?.?.(6)Pn=1(1)n n!3n2.?limn(n+1)!3(n+1)2/n!3n2=limnn+132n+1=0.?,?.4(7)Pn=2(1)n 1nsinn.?limn1nsinn/1n2=,?.(8)Pn=1(1)n+1tann(2 0)?Pn=1nn+1un?.?.?1np?nn+1?,?Pn=1un?,?,?Pn=1unnp?Pn=1nn+1un?.3.?Pn=1cosnnp(0 1?,?0 1?,?Pn=11np?.?|cosnnp|1np,?Pn=1cosnnp?.(ii)?0 x0(x x0?,?nx0 x?.?,?Pn=1annx=Pn=1annx0nx0 x?.6.?Pn=1un?,?Pn=12n1nun?.?.|2n1nun|2|un|,?Pn=1|un|?Pn=12n1n|un|?.?10.451.?.(1)Pn=1(lnx)n.?|lnx|13?,|1xnsin3n|1|x|n3n,?,?.?(,13)(13,+).2.?.(1)fn(x)=12n+x2,x +.?.limnfn(x)=0.?|fn(x)0|12n,?limn12n=0,?.(2)fn(x)=x4+en,x +.?.limnfn(x)=x4=x2.?|fn(x)x2|=enx4+en+x2enen=en2,?limnen2=0,?.(3)fn(x)=ln(1+x2n2),(a)l x +l,(b)x +.?.limnfn(x)=0.(a)?|fn(x)0|x2n2l2n2,?limnl2n2=0,?(l,l)?.(b)?xn=n,?limnf(xn)0=ln2?(,+)?.(4)fn(x)=n2x1+n2x,0 x 1.?.limnfn(x)=1.?xn=1n2,?limnf(xn)1=12?.3.?.(1)Pn=1(1)nnx2+n2,x +.?.|(1)nnx2+n2|nn2=1n3/2,?M?,?.(2)Pn=1(xnnxn+1n+1),1 x 1.?.|Pk=n+1(xkkxk+1k+1)|=|xn+1n+1|1n,limn1n=0,?.(3)Pn=1sin nx1+(x2+n2)3,x +.?.|sinnx1+(x2+n2)3|1n3,?M?,?.(4)Pn=1x1+4n4x2,x +.6?.1+4n4x2 4n2|x|,?|x1+4n4x2|14n2,?M?,?.(5)Pn=1x2(1+x)n,x +.?.Pk=n+1x2(1+x)k=x(1+x)nx1+nx1n,limn1n=0,?.(6)Pn=1sinxsinnxn2+x2,0 x 2.?.1n2+x21n,limn1n=0,?1n2+x2?0.?1n2+x2?n?,?|nPk=1sinx sinkx|=|cosx2 cosx2 cos(n+12)x|2,?,?.(7)Pn=1(1)n1x2enx2,x 0)?,?g(x)?(,+)?.?.(i)?xn=3n+1,?|Pk=n+12ksinxn3k|2n+1sin1 sin1,?(,+)?.(ii)?x M,M?,|2nsinx3n|2nx3n|M(23)n.?M?,?Pn=12nsinx3n?M,M?.(iii)|(2nsinx3n)|=|(23)ncosx3n|(23)n,?M?,?Pn=1(2nsinx3n)?(,+)?.?g(x)?(,+)?.6.?(x)=Pn=11nx?1+,+)?(0),?Pn=1lnnnx?1+,+)?(0),?(x)?(1,+)?.7?.(i)?x 1+,+?,1nx1n1+,?M?,?Pn=11nx?1+,+)?.(ii)limnlnnn/2=0,?M 0,?lnnn/2 0).limnnq1nan=1a,?(a,a).?x=a?,x=a?,?a,a).(3)Pn=0(1)nx2n+1(2n+1)(2n+1)!.limn|x2n+3|(2n+3)(2n+3)!/|x2n+1|(2n+1)(2n+1)!=0,?(,+).8(5)Pn=0 x2n+1.?,?x2 0?n N?ln1an(1+)lnn,?an 0.?Pn=1an?.?.?n N?,?ln1an(1+)lnn,?1an n1+,?an1n1+.?,?Pn=1an?.9.?y=f(x)?(x0a,x0+a)(a 0)?,?,?|f(n)(x)|M(M?).?:f(x)=Pn=01n!f(n)(x0)(x x0)n(x0 a x x0+a).?.f(x)?x=x0?Rn(x)=1(n+1)!f(n+1)(x0+(x x0)(x x0)n+1.?x (x0 a,x0+a)?,|Rn(x)|Man(n+1)!,?limnRn(x)=0,?f(x)?f(x).11