infinate series

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1、InfiniteSeriesTomRikeBerkeleyMathCircleMarch24,20021BackgroundAlthoughtherigorofthistalkwillnotbestringentenoughformathematicians,thematerialisquiteinterestingandthetechniquesquiteuseful.IdonotknowinfiniteserieswellenoughtosaythatItrulyunderstandthetopic,butIenj

2、oyinvestigatingitinthedozensofbooksoninfiniteseriesinmylibraryandrevellingintheamazingresults.ThisambivalentfeelingaboutinfiniteseriesprobablystemsfromthefactthatIdidnotunderstandanythingaboutinfiniteseriesinmycalculusclass37yearsagoandtheotherfactthatImustnowtea

3、chinfiniteserieseveryyeartomystudents.Oneofthehighlightsoftheyear(atleastforme)inmyclass,aftertheAdvancedPlacementexaminMay,istoseehowthe28yearoldEulerin1735solvedaproblemknownastheBaselProblem.ThisproblemhadbeenproposedbyJakobBernoulliin1689whenhecollectedthea

4、lloftheworkoninfiniteseriesofthe17thcenturyinavolumeentitledTractatusDeSeriebusInfinitis.Bernoulli’scommentinthisvolumewasthattheevaluation“ismoredifficultthanonewouldexpect”.Littledidheknowhowdifficultitreallywasandthatitwouldtakealmost50yearsfortheproblemtobesolve

5、d.Inmyclass,wethenseehowtheproblemcanbesolvedtothesatisfactionofmathematicianstoday,usingonlyknowledgegainedinfirstyearcalculus.ThisproblemishighlightedinthefirstvolumeofPolya’sMathematicsandPlausibleReasoning,Volume1[6]inhisdiscussionontheuseofanalogyleadingtod

6、iscovery.Itisthismethodof‘solving’theproblemthatthistalkwilladdress.Therealproofsusingelementarycalculusaremany,butnotappropriateforthiscircle.Ifyouhavestudiedcalculusthenyouwillfindproofsandreferencestoproofsin[2],[9],andanextensivebibliographyisgivenbyE.L.Sta

7、rkinMathematicsMagazine,47(1974)pp197-202.Awordofcautionisinorderaboutenteringfieldswithoutaproperfoundation.Itiseasytodothingsbyexampleandneverattainanunderstanding.ThisistheissueofHowversusWhythatPaulZeitzgaveatalkonlastyear.ItisinterestingtoseetheremarksofG.

8、Chrys-talinTextbookofAlgebra[3]writtenin1889.“Apracticehassprungupoflate(encouragedbydemandsforprematureknowledgeincertainexaminations)ofhurryingyoungstudentsintothemanipulationoft