CK12 Calculus - Single Variable

and applyingLimit Comparisontest with
.

The Root Test is inconclusiveon , but the simplerTest for Divergenceconfirmsits divergence

since always.

Example 5 is convergentforq> 0 by the AlternatingSeriesTest since

and. It is absolutelyconvergentforq> 1 by thep-test.So it is conditionallyconvergent
for 0 < q ≤ 1.

Example 6 is absolutelyconvergent by the RatioTest since
.

Example 7 Considerthe series. Noticen^2 - 10n+ 1 is never0 and is positiveforn≥
10, we couldignorethe termsbeforen= 10. Droppingthe lowerpowersofnleadsto the candidate

for applyingLimitComparisonTest since. So the series
is (absolutely)convergentby thep-test.A combinationof testsis applied.
Exercises

1. For whatvaluesofpis the series convergent?
   Test the followingseriesfor convergenceor divergence:

2.

3.

4.

5.