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1、SERIES-SUMOFASERIESDefinition1SupposewehaveaninfinitesequeneeofnumbersU123,•••”9•••Theexpressionu,+u9+u3+_.+un+..•iscalledanumericalseries.Here,thenumberui，u23,•••,u〃，•••arecalledthetermsoftheseries.Definition2Thesumofafinitenumberofterms(thefirstnterms)ofaseriesiscall
2、edthenthpartialsumoftheseries:sn=U]+u2+u3+..<+uHIfthereexistsafinitelimits=lim，itiscalledthesumoftheseries(1)andwesaythattheseriesconverges・Iflimszldoesnotexist,thenwesaythattheseries(1)divergesandhasno刃T8sum.Example1Considertheseriesa+aq+aq2+aq'+…+aq"一"+…(2)Thisisageo
3、metricprogressionwithfirsttermaandratioq(aH0)Thesumofthefirstntermsofthegeometricprogressionis(whenqH1)a-aqnaaqnQ==ntqY-q-qi-q(1)If
4、^
5、<1,thenTOasn—>8andconsequently,aaqnalimsn=lim()=川too”T8i-qi-qi-qHence,inthecaseof
6、q
7、1,thenqnT8asnTooandthenlimsndoesnotexist.Thus,when〃T8series(2)diverges.(3)Ifq=l,thentheseries(2)hastheforma+a+a+....Inthiscase,theseriesdiverges-(4)Ifq=-1,thentheseries(2)hastheforma-a+a-a+…Inthiscasewhennisevens“=0,whennisodds”=a.Thus,snhasnolimitandtheseriesdiverges・
8、Thus,ageometricprogression(whichfirsttermdifferentfromzero)convergesonlywhentheratiooftheprogressionislessthanunityinabsolutevalue.Theorem1TheconvergenceofaseriesisnotaffectedbythesuppressionofafinitenumberofitstermsoInotherwords,ifaseriesobtainedfromagivenseries(1)bys
9、uppressionofsomeofitstermsconverges,thenthegivenseriesitselfconverges.Conversely,ifagivenseriesconverges,thenaseriesobtainedfromthegivenseriesbysuppressionofseveraltermsalsoconverges.Theorem2Ifaseriesuj+u2+u3+<..+un+...convergeanditssumiss,thentheseriescu]+cu2+cu3+•••+
10、cu〃+••・wherecissomefixednumber.alsoconverges,anditssumiscs•Theorem3Iftheseriesuj+u2+u3+...+u+...andv〔+v?+v3+・••+v“+convergeandtheirsumsrespectively,aresandO,thentheseries(u,±vj)+(u2±v2)+•••+(un±v〃)•…alsoconvergeandtheirsumsares±O^respectively.Oneofthebasicquestions,whe
11、ninvestigatingseries,isthatofwhetherthegivenseriesconvergesordiverges・Weshallestablishsufficientconditionsforonetodec
