CK12 Calculus - Single Variable

10. intermediate


11. intermediate


12. challenging

Keywords

1. nondecreasingsequence


2. harmonicseries


3. geometricseries
   4.p-series


4. ComparisonTest


5. IntegralTest


6. LimitComparisonTest


7. SimplifiedLimitComparisonTest

SeriesWith Odd or EvenNegativeTerms

LearningObjectives

• Demonstratean understandingof alternatingseries


• Applythe AlternatingSeriesTest to an appropriateseries


• Explainthe differencebetweenabsoluteand conditionalconvergence


• Determineabsoluteand/orconditionalconvergenceof series

AlternatingSeries(harmonic,geometric,p-series)
Alternatingseriesare serieswhosetermsalternatebetweenpositiveand negativesigns.Generally, alter-
natingserieslook like one of theseexpressions:
u 1 - u 2 +u 3 - u 4 + ... or -u 1 +u 2 - u 3 +u 4 - .....

Eitherthe termswith the evenindicescan havethe negativesign or the termswith the odd indicescan have
the negativesign.The actualnumbersrepresentedby theui'sare positive.

Thereare severaltypesof alternatingseries.One type is thealternatingharmonicseries:

.

This serieshas termsthat look like the harmonicseriesbut the termswith evenindiceshavea negative
sign.
Anotherkind is thealternatinggeometricseries.Hereis one example: