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If limkakbk=andk=1ak diverges, explain why we cannot draw any conclusions about the behavior ofk=1bk.

Short Answer

Expert verified

Thebehavioroftheseriesbkk=1cannotbedeterminedwhenlimkakbk=andakk=1isdivergentholdsbecausetheseriesakk=1mayormaynotbedivergent,eventhoughlimkakbk=holds.

Step by step solution

01

Step 1. Given information

limkakbk=andk=1akdiverges

02

Step 2. Finding the value of limit 

Consider the convergent series k=1ak=k=11kand the series k=1bk=1k2.

Thevalueoflimkakbkis:limkakbk=limk1k1k2=limkk=

03

Step 3. Result

Theseriesbkk=1=1k2isconvergentbuttheseriesakk=1=1kisdivergent.Thebehavioroftheseriesbkk=1cannotbedeterminedwhenlimkakbk=andakk=1isdivergentholdsbecausetheseriesakk=1mayormaynotbedivergent,eventhoughlimkakbk=holds.

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