数学分析课后习题答案--高教第二版(陈纪修)--10章

10.11.Sn(x)=,(i)xnx−e∈)1,0((ii)x∈),1(+∞Sn(x)=x,xnx−e∈),0(+∞Sn(x)=sinnx,(i)x∈),(+∞−∞(ii)x∈],[AA−()0ASn(x)=arctannx,(i)x∈)1,0((ii)x∈),1(+∞Sn(x)=221nx+,x∈),(+∞−∞Sn(x)=nx(1x)n,x∈]1,0[Sn(x)=nxlnnx,(i)x∈)1,0((ii)x∈)),1(+∞Sn(x)=nnxx+1,(i)x∈)1,0((ii)x∈),1(+∞Sn(x)=(sinx)n,x∈],0[πSn(x)=(sinx)n1,(i)x∈[0,]π(ii)x∈],[0δδπδ−Sn(x)=nnx⎟⎠⎞⎜⎝⎛+1(i)x∈),0(+∞(ii)x∈],0(A()0ASn(x)=⎟⎟⎠⎞⎜⎜⎝⎛−+xnxn1,(i)x∈),0(+∞,(ii)[)0,,+∞∈δδx1(i)0)(=xS)()(sup),()1,0(xSxSSSdnxn−=∈1=/0∞→n{}()nSx(0,1)(ii)0)(=xS)()(sup),(),1(xSxSSSdnxn−=+∞∈ne−=)(0∞→→n{}()nSx(1,)+∞20)(=xS)()(sup),(),0(xSxSSSdnxn−=+∞∈ne1=)(0∞→→n1{}()nSx(0,)+∞3(i)0)(=xS)()(sup),(),(xSxSSSdnxn−=+∞−∞∈1=/0∞→n{}()nSx(,)−∞+∞(ii)0)(=xSπAn2)()(sup),(],[xSxSSSdnAAxn−=−∈nA≤)(0∞→→n{}()nSx[,]AA−4(i)2)(π=xS)()(sup),()1,0(xSxSSSdnxn−=∈2π=/0∞→n{}()nSx(0,1)(ii)2)(π=xS)()(sup),(),1(xSxSSSdnxn−=+∞∈narctan2−=π)(0∞→→n{}()nSx(1,)+∞5xxS=)(nxnxxSxSn11)()(22≤−+=−)()(sup),(),(xSxSSSdnxn−=+∞−∞∈)(0∞→→n{}()nSx(,)−∞+∞60)(=xS=−)1()1(nSnSnnn)11(−/0∞→n{}()nSx[0,1]7(i)0)(=xS0)0()0(=+−+SSn2[]=−)()(xSxSdxdn0)ln1(1+nxn)2(≥nnnxSxSSSdnxnln)()(sup),()1,0(=−=∈)(0∞→→n{}()nSx(0,1)(ii)0)(=xS=−)2()2(nSnSn2ln2/0∞→n{}()nSx(1,)+∞8(i)0)(=xS=−−−)11()11(nSnSnnnnn)11(1)11(−+−/0∞→n{}()nSx(0,1)(ii)1)(=xS=+−+)11()11(nSnSn1)11(1)11(−+++nnnn/0∞→n{}()nSx(1,)+∞9⎪⎪⎩⎪⎪⎨⎧≠∈==2],,0[021)(πππxxxxS],0[π∈nxnxn11sin−=2π≠nx=−)()(nnnxSxSnn)11(−/0∞→n{}()nSx[0,]π10(i)⎩⎨⎧==ππxxxS01,00)(),0(π∈nxnnx21sin=3=−)()(nnnxSxS121−/0∞→n{}()nSx(0,)π(ii)1)(=xS)()(sup),(xSxSSSdnn−=],[x−∈δπδδn1sin1−=)(0∞→→n{}()nSx[,]δπδ−11(i)xexS=)(=−)()(nSnSnnne−2/0∞→n{}()nSx(0,)+∞(ii)xexS=)(0)0()0(=+−+SSnn[]=−)()(xSxSdxdn011−⎟⎠⎞⎜⎝⎛+−xnenx)()(sup),(],0(xSxSSSdnAxn−=∈nAnAe⎟⎠⎞⎜⎝⎛+−=1)(0∞→→n{}()nSx(0,]A12(i)xxS21)(==−)1()1(nSnSnn⎟⎠⎞⎜⎝⎛−232/0∞→n{}()nSx(0,)+∞(ii)xxS21)(=Sn(x)=⎟⎟⎠⎞⎜⎜⎝⎛−+xnxn1)(2111xSxxnx=++=4[]041)1()1(21)()(23++++−=−xnxxnxxxSxSdxdn)()(sup),(),[xSxSSSdnxn−=+∞∈δ)()(δδSSn−=δδδ211+⎟⎟⎠⎞⎜⎜⎝⎛−+−=nn)(0∞→→n{}()nSx[,)δ+∞2.Sn(x)=n(nxnx2){S(x)}n]1,0[∞→nlim∫10)(xSndx≠∫∞→10limnSn(x)dx{Sn(x)}]1,0[0)(=xSnxn11−==−)()(nnnxSxS+∞→⎥⎦⎤⎢⎣⎡−−−nnnnn2)11()11({Sn(x)}]1,0[∞→nlim∫10)(xSndx∞→=nlimxxxnnnd)(102∫−21=S∫∞→10limnn(x)dx0=dx∞→nlim∫10)(xSn≠∫∞→10limnSn(x)dx3.Sn(x)=221xnx+{Sn(x)}),(+∞−∞⎭⎬⎫⎩⎨⎧)(ddxSxn),(+∞−∞∞→nlimxddSn(x)=xdd∞→nlimSn(x)x∈),(+∞−∞1Sn(x)=221xnx+0)(=xSnxnxxSxSn211)()(22≤+=−)(0∞→→n5{Sn(x)}),(+∞−∞2)(xSdxdn22222)1(1xnxn+−=)(lim)(xSdxdxnn∞→=σ⎩⎨⎧≠==0001xxnxn21=)(nnxSdxd2512)(=−nxσ/0∞→n⎭⎬⎫⎩⎨⎧)(ddxSxn),(+∞−∞30=xxdd∞→nlimSn(x)0=)(lim)(xSdxdxnn∞→=σ1=0=x∞→nlimxddSn(x)=xdd∞→nlimSn(x)4.Sn(x)=n1arctanxn{Sn(x)}),0(+∞∞→nlimxddSn(x)=xdd∞→nlimSn(x)?Sn(x)=n1arctannxnnnxxxS21'1)(+=−=)(xS∞→nlimSn(x)0=0)('=xS)1('21)1(lim'SSnn≠=∞→∞→nlimxddSn(x)=xdd∞→nlimSn(x)1=x5.Sn(x)=aa{Snxxen−αn(x)}]1,0[∞→nlim∫10)(xSndx=S∫∞→10limnn(x)dxx∈[0,1]∞→nlimxddSn(x)=xdd∞→nlimSn(x)6(1)S=)(xS∞→nlimn(x)0==)('xSn0)1(=−−nxennxαnx1==−=∈)()(sup),(]1,0[xSxSSSdnxn11)1(−−=ennSnα0),(lim=∞→SSdnn1α1α{Sn(x)}]1,0[(2)S∫∞→10limnn(x)dx∫==100)(dxxS∫=10)(dxxSnnennn−−−+−)11(12αα2α.∞→nlim∫10)(xSndx=S∫∞→10limnn(x)dx(3)xdd∞→nlimSn(x)xdd=0)(=xSxddSn(x))1(nxennx−=−α)1(limnxenxn−−∞→⎩⎨⎧=∈=01]1,0(0xx0α∞→nlimxddSn(x)=xdd∞→nlimSn(x)x∈[0,1]6.S'(x)),(baSn(x)=⎥⎦⎤⎢⎣⎡−⎟⎠⎞⎜⎝⎛+)(1xSnxSn{Sn(x)}S'(x)),(baS∞→nlimn(x))('xS=0∀η,{})(xSn[]ηη−+ba,)('xSηα0,)('xS[]αα−+ba,,0,0∃∀δε,∈∀,'xx[]αα−+ba,,δ−'xx,ε−)(')'('xSxS⎭⎬⎫⎩⎨⎧⎥⎦⎤⎢⎣⎡−⎥⎦⎤⎢⎣⎡=αηδ1,1maxN,Nn∈x[]ηη−+ba,∈+nx1[]αα−+ba,,7=−)(')(xSxSnεξ−)(')('xSS{Sn(x)}S'(x)),(ba7.)(0xS],0[aSn(x)=dtn=∫−xntS01)(,2,1{Sn(x)}0],0[aMxS≤)(0,MxdttSxSx≤=∫001)()(∫∫≤=xxMtdtdttSxS0012)()(!22xM=∫∫=−≤=−−xnnxnnnxMdtntMdttSxS0101!)!1()()(!!naMnxMnn≤0)!(lim=∞→naMnn{Sn(x)}0],0[a8.S(x)S(1)=0{x]1,0[nS(x)}[0,1]S(x)]1,0[MxS≤)(,0)1(=S0,0∃∀δε,[]1,1δ−∈∀xε)(xSxn{}nx[]δ−1,0,N∃,Nn∀[]δ−∈∀1,0xMxnεε)(xSxn8{x]1,0[∈xnS(x)}[0,1]910.21.x[0,1]∑∞=−0)1(nnxx∑∞=−02)1(nnxxx[0,1]x∑∞=−032ennxx[)+∞,0(i)x∑∞=−02ennxx[)+∞,0(ii)x[)+∞,δ0∑∞=+0231nxnxx(,)∑∞=+1344sinnxnnxx(,)x[0,1]∑∞=−−0)1()1(nnnxx∑∞=+−12)1(nnxnx(,)∑∞=031sin2nnnx(i)x(0,)(ii)x[)+∞,δ0∑∞=1sinsinnnnxxx(,)∑∞=+022)1(nnxxx(,)∑∞=+−022)1()1(nnnxxx(,)1∑=−=nkknxxxS0)1()(11+−=nx{}1+nx∑]1,0[∞=−0)1(nnxx]1,0[2nnxxxu2)1()(−=]1,0[)2()(0+≤≤nnuxunn2)2(4+n∑∞=+02)2(4nnWeierstrass∑∞=−02)1(nnxx]1,0[1323)(nxnexxu−=1≥n),0[+∞)23()(0nuxunn≤≤23nK=23463−=eK∑∞=023nnKWeierstrass∑∞=−032ennxx),0[+∞4(i)N2)(nxnxexu−=)(2Nnnm=nxn1=),0[+∞∈=∑+=mnknkxu1)(++−2)1(nxnnex+++−2)2(nxnnex−22nnxnex22nnxnenx−+∞→=−2en)(∞→n∑∞=−02ennxx∑∞=−02ennxx),0[+∞(ii)2)(nxnxexu−=221δn)(xunx),[+∞δnnexu2)(0δδ−≤≤Weierstrass∑∞=−02nneδδ∑∞=−02ennxx),[+∞δ5231)(xnxxun+=1≥n2321)(nxun≤∑∞=02321nnWeierstrass∑∞=+0231nxnx),(+∞−∞6344sin)(xnnxxun+=1≥n341)(nxun≤∑∞=0341nn2Weierstrass∑∞=+1344sinnxnnx),(+∞−∞7nnxxxa)1()(−=nnxb)1()(−={})(xan]1,0[∈xn]1,0[1)(0≤∑=nkkxbDirichlet∑∞=−−0)1()1(nnnxx]1,0[821)(xnxan+=nnxb)1()(−={})(xan),(+∞−∞∈xn),(+∞−∞1)(1≤∑=nkkxbDirichlet∑∞=+−12)1(nnxn),(+∞−∞9(i)xxunnn31sin2)(=πnnx32=),0(+∞∈+∞→=nnnxu2)({})(xun),0(+∞∑∞=031sin2nnnx),0(+∞(ii)xxunnn31sin2)(=),[+∞∈δxnnxu⎟⎠⎞⎜⎝⎛≤321)(δnn⎟⎠⎞⎜⎝⎛∑∞=3210δWeierstrass∑∞=031sin2nnnx),[+∞δ10nxan1)(=nxxxbnsinsin)(=)(xanx3{})(xan),(+∞−∞∈xn),(+∞−∞=∑=nkkxb1)(∑=nkkxxx1sin2sin22cos22cos)21cos(2cos≤−+⋅=xxnxDirichlet∑∞=1sinsinnn