[加上均匀收敛(uniform

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1、Chaper7InfiniteseriesoffunctionsSection7.1UniformConvergenceofSequenceDefinition.Letandlet.(1)..(2).Definition7.1.Let,.WesaythatconvergestofifNotation:Remark..Remark7.3.Notingthatdiscontinuous.Remark7.4.LetForexamples,NotingthatRemark7.6:(3)Proof.Then.Questiondiffere

2、ntiableIsfdifferentiable?Remark7.3isdifferentiabledifferentiableat1.Remark7.5:differentiableBut<結論>differentiableIsfdifferentiable?Nodifferentiableandisfdifferentiable?Nointegrableand[加上均勻收斂(uniformconvergence)才會對]Definition7.7:Let,.WesaythatconvergesuniformlytofifNo

3、tation:Example7.8:Proof.ExsmpleProof.Theorem7.9:.Proof...,Wehave.Remark:andTheorem7.10.ThenProof.(1)..ForthisP,(2)Lemma7.11:,ifandonlyif.Proof.Theorem7.12:Proof.differentiabledifferentiableHence,by(*)andExercise3(c),.(2)differentiable.Ontheotherhand,(3)Repeatingthepr

4、oduceof(1)and(2)Weknowthatdifferentiableand.