进一步探讨正项级数的收敛与发散

:10075570(2002)02-0021-03X(,550008):,,,:;;:O173:AFurtherstudyontheconvergenceanddivergenceofserriesofpositivetermsFENGLin2an(Departmentofmathematics,GuiyangJuniorTeachersCollege,Guiyany,Guizhou550008,China)Abstract:Thispaperintroducesadiscriminantfortheconvergenceanddivergenceofseriesofposi2tiveterms.Theconvergenceanddivergenceofsomekeyserieshavebeenprovedtoexplainthatnei2theroneoftheseriesofpositivetermsistheslowestdivergencenoroneofthemisthelowestconver2gence.Keywords:seriesofpositiveterms;generalizedharmonicseries;comparativediscriminant0,,1n=1an(an0),Sn=a1+a2++an,n=1anSpnpF1,p1nkan+1Sn+1+an+2Sn+2++an+kSn+kan+1+an+2+an+kSn+k=1-SnSn+kn=1an,nv,Sn+,,n,kSnSn+k12an+1Sn+1+an+2Sn+2++an+kSn+k12n=1anSn,,pF1,n=1anSpn12X:2002-02-1620220025()JournalofGuizhouNormalUniversity(NaturalSciences)Vol.20.No.2May2002p1,f(x)=1xp-1,[Sn-1,Sn]1p-1(1Sp-1n-1-1Sp-1n)=anpnn(Sn-1,Sn)anSpnanpn=1p-11Sp-1n-1-1Sp-1nSp-1n+,n=11p-1(1Sp-1n-1-1Sp-1n),n=1anSpnp121)n=11n,,n=11n(1+12+13++1n)p(1)pF1,p1(1)n=21n(lnn)p(Euler)Hn=1+12+13++1nlnn+c+nc=0.57721,nv0(nv+)1+12+13++1nlnn1n(1+12+13+1n)p1n(lnn)p(p0),pF1,n=21(nlnn)pn,Hn=lnn+c+n2lnn1n(1+12+13+1n)p1n(2lnn)p=12p1n(lnn)p(p0),p1,n=21n(lnn)pn=21n(lnn)p(1)n=1nn!n2:n1a1+1a2++1anFna1a2anFa1+a2++ann(2)ai0i=1,2,,n:11+12+13+1nFnn!n1n1+12+13+1nFnn!n2(1),n=2nn!n2(2)n=21nlnn,,n=21nlnn(12ln2+13ln3++1nlnn)p(3)pF1,p1(3)n=31nlnn(lnlnn)pf(x)=lnlnx,,[n,n+1]1(n+1)ln(n+1)lnln(n+1)-lnlnn1nlnn(nE2)13ln3+14ln4++1(n+1)ln(n+1)lnln(n+1)-lnln212ln2+13ln3++1nlnn(4)12ln2+13ln3++1nlnnlnln((n+1)-lnln2lnln(n+1)lnlnn1nlnn(12ln2+13ln3++1nlnn)p1nlnn(lnlnn)p(p0,nE3),pF1,n=31nlnn(lnlnn)p(4)12ln2+13ln3++1nlnnlnln(n+1)-lnln2+12ln22lnlnn+lnlnn=3lnlnn22()201nlnn(12ln2+13ln3++1nlnn)p1nlnn(3lnlnn)p=13p1nlnn(lnlnn)p(p1),p1,n=31nlnn(lnlnn)pn=31nlnn(lnlnn)pn=21n(lnn)p,n=k1nlnnlnlnn(lnlnlnn)ppF1,p1,n=k1nlnnlnlnn(lnlnlnn)ppF1,p1(3)n=2nn!ln(n!)n3lnn(2)112ln2+13ln3++1nlnnFnn!ln2ln3lnnn=nn!nnln2ln3lnnFnn!nln2+ln3++lnnn=nn!lnn!n21nlnn(12ln2+13ln3++1nlnn)Fnn!ln(n!)n3lnn(3),n=2nn!ln(n!)n3lnn,n=1an(an0),Sn=a1+a2++an,n=kanSn(lnSn)pn=knSnlnSn(lnlnSn)pn=kanSnlnSnlnlnSn(lnlnlnSn)ppF1,p1n=kanSn(lnSn)p,n=1anSn,n=1anSn(a1S1+a2S2++anSn)ppF1,p1f(x)=lnx,[Sn-1,Sn]lnSn-lnSn-1=annn(Sn-1,Sn)anSnlnSn-lnSn-1anSn-1a2S2+a3S3++anSnlnSn-lnS1a2S1+a3S2++anSn-1a1S1+a2S2++anSnlnSn-lnS1+a1S12lnSn(n)anSn(a1S1+a2S2++anSn)panSn(2lnSn)p=12panSn((lnSn)p(p0)p1,n=kanSn(lnSn)panSnanSn-1,n=2anSn-1n=2anSn-1(a2S1+a3S2++anSn-1)ppF1,p1a2S1+a3S2++anSn-1lnSn-lnS112lnSnanSn-1(a2S1+a3S2++anSn-1)pan12Sn(12lnSn)p=2p+1anSn(lnSn)p(p0)pF1,n=kanSn(lnSn)p:[1],.[M].:,1998.[2],.[M].:,1983.322: