Hint: and existsand equalse.
Summaryof Proceduresfor DeterminingConvergence
We haveseenvarioustest for convergenceof in action.To summarize,the key phraseis "recognize
the formofan".
Someseriesare hard for any test evenif it works.However, if a specifictest workswellsfor a series,we
shoulddevelopthe insightto pick it. So applyingthe list of testsin a fix orderis not advised.Nonetheless,
sometests are relativelysimpleto applyif we haveno idea towardsa seriesat all. The followingis a summary:
Test Formofan Comments
1 No/littletest geometric,harmonic,p-test clearanswer
2 Test of divergence inexpensivetest
3 IntegralTest corresp.integralin nice closedform easyintegration
4 AlternatingSeries checkconditionsonbn
5 (Limit)Comparison needcompanionknownseries compare
evaluateratio
recognizegoodform(^6) RatioTest
(^7) RootTest anresemblescnn evaluatenthroot
8 Combination compositeof forms combinedmethods
*: the inconclusivecasesneedothertests.
Example 1
divergesby the IntegralTest since diverges.
Example 2 n diverges by Limit Comparison Test (against the harmonic series) since
and the harmonicseriesdiverges.
Example 3 converges with Limit Comparison Test (against ) since
and the latterseriesconverges.
Example 4 Convergenceof is determinedwith by the RootTest. For example
,convergessince ,but (noexponent
n) divergesby rationalizingthe numerator: