CK12 Calculus - Single Variable

Thus. Then. This tells us that the absolute
valueof the erroror remainderis less than 0.13.

Absoluteand ConditionalConvergence
Thereare othertypesof convergencefor infiniteseries:absoluteconvergenceandconditionalconver-
gence.

AbsoluteConvergenceLet be an infiniteseries.Thenthe

seriesisabsolutelyconvergentif converges.

The infiniteseries is the seriesmadeby takingthe ab-

solutevaluesof the termsof the series
The convergenceof the seriesof absolutevaluestells us somethingaboutthe convergenceof the series.

Theorem If , then

also converges.

This tells us that if you can showabsoluteconvergence,then the seriesconverges.
If the seriesof absolutevaluediverges,we cannotconcludeanythingaboutthe series.
Example 3

Determineif the series convergesabsolutely.
Solution

We find the seriesof absolutevalues: , whichbehaveslike the series

. This is ap-serieswithp= 4. The series convergesabsolutelyand henceconverges.
Example 4