欢迎来到天天文库
浏览记录
ID:34665002
大小:92.05 KB
页数:6页
时间:2019-03-08
《infinate series》由会员上传分享,免费在线阅读,更多相关内容在学术论文-天天文库。
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
此文档下载收益归作者所有