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Let 0 < p < 1. Evaluate the limitlimk1/klnk1/kp

Explain why we cannot use a p-series with 0 < p < 1 in a limit comparison test to verify the divergence of the seriesk=21klogk

Short Answer

Expert verified

Thep-seriesisdivergentfor0<p<1.Thelimitcomparisontestiftheratioiszerostatesthatiflimkakbkandk=1bkconverges,thenk=1akconvergesThelimitcomparisontestfailstogiveanyinformationaboutthedivergenceoftheseriesk=21klnk

Step by step solution

01

Step 1. Given

k=21klogk

02

Step 2.Finding the value of the expression

Thevalueoftheexpressionislimk1/klnk1/kplimk1/klnk1/kp=limkkpklnk=limkkp-1lnk=limk(p-1)kp-21k(using'slhospitalrule)=limk(p-1)k1-p=0

03

Step 3. Limit comparison test

ThelimitcomparisonteststatesthatforandbetwoserieswithpositivetermsthenIflimkakbk=LWhereLisanypositiverealnumbertheneitherbothconvergesordiverges.Iflimkakbk=0andk=1bkconvergesthenk=1akconverges.Iflimkakbk=andk=1bkdivergesthenk=1akdiverges.

04

Step 4. Result

Thevalueoftheexpressionislimk1/klnk1/kp=0Thep-seriesisdivergentfor0<p<1.Thelimitcomparisontestiftheratioiszerostatesthatiflimkakbkandk=1bkconverges,thenk=1akconvergesThelimitcomparisontestfailstogiveanyinformationaboutthedivergenceoftheseriesk=21klnk

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