函数项级数的一致收敛性

世界观人人都有，而哲学只有经过系统的学习的人才能掌握它。世界观是自发成的，是不系统的、不自觉的、缺乏严密的逻辑和理论论证，而哲学则是把自发的、零散的、朴素的世界观加以理论化和系统化，因而具有严密的逻辑和完整的理论体系。appearanceoftheweldappearancequalitytechnicalrequirementsoftheprojectmustnothaveamoltenmetalstreamdoesnotmeltthebasemetaltoweld,weldseamandheat-affectedzonesurfacemustnothavecracks,pores,defectssuchascraterandash,surfacesmoothing,weldandbasemetalshouldbeevenlysmoothtransition.Width2-3mmfromtheedgeofweldGroove.Surfacereinforcementshouldbelessthanorequalto1+0.2timestheslopeedgewidth,andshouldnotbegreaterthan4mm.Depthofundercutshouldbelessthanorequalto0.5mm,totallengthoftheweldsonbothsidesundercutnotexceed10%oftheweldlength,andlongcontinuousshouldnotbegreaterthan100mm.Wrongsideshouldbelessthanorat0.2T,andshouldnotbegreaterthan2mm(wallthicknessmmt)incompleteornotallow7.57.5.1installationqualityprocessstandardsoftheelectricalenclosureCabinetsurfaceisclean,neat,nosignificantphenomenonofconvex,closetonature,closethedoor.7.5.2CabinetCabinetfacepaintsnopaint,returnedtorusted,consistentcolor.7.5.3uniformindirectgapfromtoptobottom,slotwidth1.5mm7.5.4adjacentCabinetsurfaceroughnessis0.7.5.5thecabinetsfirmlyfixed,craftsbeautiful.7.5.6Cabinetsurfacegauge,switchcabinetmarkclear,neat,firmpaste.7.5.7Terminalrowofneat,isreliable,theappearanceiscleanandnotdamaged.7.5.8cablesneatandclean,solidbinding,bindingprocessinappearance.7.5.9thefirstcableproductionfirm,craftsbeautiful,clearsignagedoesnotfade.7.5.10fireproofpluggingtight,nocracksandpores.7.67.6.1ofthestandardelectricalwiringqualitytechnologycablea,themulti-corewirebuncharrangementshouldbeparalleltoeachother,horizontalwireharnessorwireshouldbeperpendiculartothelongitudinalmulti-corewirebunch.Thedistancebetweenthewireharnessandwireharnesssymmetry,andascloseaspossible.B-corewiringharnessintoround,multi-corewirebunchusedgwirebinding,fastening第三节函数项级数的一致收敛性本节将讨论函数项级数有关性质。定义1设)(1xu，)(2xu，……，)(xun，……，是集合E上的函数列，我们称形为)(1xu+)(2xu+……+)(xun+……为E上的函数项级数，简记为1)(nnxu。其中)(xun称为第n项.)(xuk+)(1xuk+……+)(xun+……也记为knnxu)(.记号中n可以用其它字母代之.同研究常数项级数一样，我们类似可以定义其收敛性。定义2设1)(nnxu是集合E上的函数项级数，记niinxuxS1)()(=)(1xu+)(2xu+……+)(xun，它称为级数1)(nnxu的部分和函数（严格地说是前n项部分和函数）.)(xSn称为1)(nnxu的部分和函数列。如果)(xSn在0x点收敛，我们也说1)(nnxu在0x点收敛或称0x为该级数的收敛点。如果|)(|1nnxu在0x点收敛，我们称1)(nnxu在0x点绝对收敛。非常容易证明绝对收敛一定收敛。)(xSn的收敛域也称为该级数的收敛域。如果)(xSn在0x点不收敛，我们说1)(nnxu在0x点发散。如果)(xSn在D上点态收敛于)(xS，我们称1)(nnxu在D上点态收敛于)(xS.)(xS称为该级数的的和函数。)()()(xSxSxRnn称为该级数关于前n项部分和的余项.世界观人人都有，而哲学只有经过系统的学习的人才能掌握它。世界观是自发成的，是不系统的、不自觉的、缺乏严密的逻辑和理论论证，而哲学则是把自发的、零散的、朴素的世界观加以理论化和系统化，因而具有严密的逻辑和完整的理论体系。appearanceoftheweldappearancequalitytechnicalrequirementsoftheprojectmustnothaveamoltenmetalstreamdoesnotmeltthebasemetaltoweld,weldseamandheat-affectedzonesurfacemustnothavecracks,pores,defectssuchascraterandash,surfacesmoothing,weldandbasemetalshouldbeevenlysmoothtransition.Width2-3mmfromtheedgeofweldGroove.Surfacereinforcementshouldbelessthanorequalto1+0.2timestheslopeedgewidth,andshouldnotbegreaterthan4mm.Depthofundercutshouldbelessthanorequalto0.5mm,totallengthoftheweldsonbothsidesundercutnotexceed10%oftheweldlength,andlongcontinuousshouldnotbegreaterthan100mm.Wrongsideshouldbelessthanorat0.2T,andshouldnotbegreaterthan2mm(wallthicknessmmt)incompleteornotallow7.57.5.1installationqualityprocessstandardsoftheelectricalenclosureCabinetsurfaceisclean,neat,nosignificantphenomenonofconvex,closetonature,closethedoor.7.5.2CabinetCabinetfacepaintsnopaint,returnedtorusted,consistentcolor.7.5.3uniformindirectgapfromtoptobottom,slotwidth1.5mm7.5.4adjacentCabinetsurfaceroughnessis0.7.5.5thecabinetsfirmlyfixed,craftsbeautiful.7.5.6Cabinetsurfacegauge,switchcabinetmarkclear,neat,firmpaste.7.5.7Terminalrowofneat,isreliable,theappearanceiscleanandnotdamaged.7.5.8cablesneatandclean,solidbinding,bindingprocessinappearance.7.5.9thefirstcableproductionfirm,craftsbeautiful,clearsignagedoesnotfade.7.5.10fireproofpluggingtight,nocracksandpores.7.67.6.1ofthestandardelectricalwiringqualitytechnologycablea,themulti-corewirebuncharrangementshouldbeparalleltoeachother,horizontalwireharnessorwireshouldbeperpendiculartothelongitudinalmulti-corewirebunch.Thedistancebetweenthewireharnessandwireharnesssymmetry,andascloseaspossible.B-corewiringharnessintoround,multi-corewirebunchusedgwirebinding,fastening2)(xRn称为该级数的余项函数列.如果)(xSn在D上一致收敛于)(xS，我们称1)(nnxu在D上一致收敛于)(xS，或1)(nnxu在D上一致收敛.如果)(xSn在D上内闭一致收敛于)(xS，我们称1)(nnxu在D上内闭一致收敛.用N的进行叙述将是：设1)(nnxu是D上函数项级数，)(xS是D上函数。若对任意0，总存在一个正数正数N（只能依赖于，绝对不依赖于x），当Nn时，对一切的Dx，总有|)()(|1xSxunii，则称该函数项级数在D上一致收敛于)(xS.同样一致收敛一定点态收敛.例1定义在（—∞，+∞）上的函数项级数（几何级数）nnnxxxx2111的部分和函数是xxxSnn11)(.显然当|x|1时xxSnn11)(lim.1||x时，几何级数是发散的。其收敛域是（—1，1）.显然几何级数在（—1，1）上不是一致收敛的.函数列的有关结论，都可以不加证明地推广到函数项级数.定理11.8（函数项级数一致收敛Cauchy准则）函数项级数1)(nnxu在集合D上一致收敛的充分必要条件是：对任意ε0,总存在正数N，使得当正整数m，n，有mnN时，对一切的x∈D,都有。.|)()()(|11xuxuxumnn世界观人人都有，而哲学只有经过系统的学习的人才能掌握它。世界观是自发成的，是不系统的、不自觉的、缺乏严密的逻辑和理论论证，而哲学则是把自发的、零散的、朴素的世界观加以理论化和系统化，因而具有严密的逻辑和完整的理论体系。appearanceoftheweldappearancequalitytechnicalrequirementsoftheprojectmustnothaveam