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PDF-Math3210-3HW25SolutionsConvergenceTests1.Determinethevaluesofpforwhich PDF document

nlognpconvergesProofWewillusetheintegraltestLetfx1 xlogxpSowewanttocomputeZ12fxdxLetsmakethesubstitutionulogxsodu1 xdxsoourintegralbecomesZ1 xupxduZupduTherearetwocasestoconsider

n(logn)pconverges.Proof:Wewillusetheintegraltest.Letf(x)=1 x(logx)p.SowewanttocomputeZ12f(x)dx.Let'smakethesubstitutionu=logxsodu=1 xdxsoourintegralbecomesZ1 xupxdu=Zupdu.Therearetwocasestoconsider:

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