CK12 Calculus - Single Variable

6. challenging


7. challenging


8. challenging


9. challenging


10. intermediate


11. challenging


12. intermediate

Keywords
alternatingseries
alternatingharmonicseries
alternatinggeometricseries
alternatingp-series
AlternatingSeriesTest
AlternatingSeriesRemainder
conditionalconvergence
absoluteconvergence
rearrangement

RatioTest, RootTest, and Summaryof Tests

RatioTest
We haveseenthe IntegralTest,(Limit)ComparisonTest and AlternatingSeriesTest whichimposeconditions
on the sign ofan. We are goingto introducetwo testsfor a strongerversionof convergencethat do not.

Definition(Absoluteconvergence)

A series isabsolutelyconvergentif the seriesof the absolutevalues is convergent.
To this end, we needto distinguishthe othertype of convergence.
Definition(Conditionalconvergence)

A series isconditionallyconvergentif the seriesis convergentbutnotabsolutelyconvergent.
TheoremIf