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Use the divergence test to prove that a p-seriesk=11kp diverges when p < 0.

Short Answer

Expert verified

Theexpression1kpiszeroforvaluesofpwhichareequaltoorgreaterthan1.Tomaketheexpression1kpnon-zero,thevalueofpmustbelessthan0.Thus,bydivergencetest,theseriesk=11kpisdivergentforp<0.

Step by step solution

01

Step 1. Given information is:

k=11kp

02

Step 2. Finding the limit:

Thevalueofthesequence{ak}=1kpis:limkak=limk1kpTheexpression1kpiszeroforvaluesofpwhichareequaltoorgreaterthan1.Theseriesk=11kpisdivergentonlywhenlimkakdoesnotconvergetozero.

03

Step 3. Result

Tomaketheexpression1kpnon-zero,thevalueofpmustbelessthan0.Thus,bydivergencetest,theseriesk=11kpisdivergentforp<0.

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