(B) If or , then the seriesis absolutelydivergent.
(C) If , then the test is inconclusive.
The proofis similarto that of the RatioTest and is left as an exercise.
Example 1 Consider wherep> 0. We alreadyknowit is convergentexactlywhenp> 1.
To applythe RootTest, we need whichis 1 after somework.Alternatively, we couldcheck
so by the argumentsimilarto the proofof the RatioTest. The Root
Test is also inconclusive.
Example 2 Test the series for convergence.
Solution.Let. Then. So the seriesis absolutely
absolutelyconvergent.
Whatif we haveappliedthe RatioTest?
We couldstill arguethe limit is with somework.So we shouldlearnto applythe right test.
Exercise
Determinewhetherthe followingseriesis absolutelyconvergent,conditionallyconvergent,or divergentwith
the RootTest and othertestsif necessary:
1.
2.
3.
- If , then.
- (Hard)Is the series convergent?