9.4ConvergenceTests
DivergenceTestInstatinggeneralresultsaboutconvergenceordivergenceofseries,itisconvenienttousethenotationasagenerictemplateforaseries,thusavoidingtheissueofwhetherthesumbeginswithk=0ork=1orsomeothervalue.Indeed,wewillseeshortlythatthestartingindexvalueisirrelevanttotheissueofconvergence.Thekthterminaninfiniteseriesiscalledthegeneraltermoftheseries.
DivergenceTestThefollowingTheoremestablishesarelationshipbetweenthelimitofthegeneraltermandtheconvergencepropertiesofaseries.
DivergenceTestThealternativeformofpart(a)givenintheprecedingTheoremissufficientlyimportantthatwestateitseparatelyforfuturereference.
Note:isnotsufficientconditionforconvergenceForexample,Eventhoughbutitisdivergent.Infact,butItiscontradiction.
Example1Doesthefollowingseriesconvergeordiverge?
Example1Doesthefollowingseriesconvergeordiverge?Itdivergessince
Example:DeterminetheconvergenceofthesequencesSolution:(1)LetthenSohenceThenitisdivergent.
Since(2)
AlgebraicPropertiesofInfiniteSeries9.4.3Theorem
AlgebraicPropertiesofInfiniteSeries9.4.3Theorem
Example2Findthesumoftheseries
Example2FindthesumoftheseriesThefirstisaconvergentgeometricserieswitha=?,r=?
Example2FindthesumoftheseriesThefirstisaconvergentgeometricserieswitha=?,r=?Thesecondisalsoaconvergentgeometricwitha=2,r=1/5
Example3Determinewhetherthefollowingseriesconvergeordiverge.
Example3Determinewhetherthefollowingseriesconvergeordiverge.Thefirstseriesisaconstanttimesthedivergentharmonicseriessoitdiverges.
Example3Determinewhetherthefollowingseriesconvergeordiverge.Thefirstseriesisaconstanttimesthedivergentharmonicseriessoitdiverges.Thesecondresultsbydeleting
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