.
The odd-indexedtermsof this serieshavethe negativesign.
Thealternatingp-seriesis anothertype of alternatingseries.An examplecouldlook like this:
Fromall of theseexamples,we can see that the alternatingsignsdependon the expressionin the power
of –1 in the infiniteseries.
The AlternatingSeriesTest
As its nameimplies,theAlternatingSeriesTestis a test for convergencefor serieswho havealternating
signsin its terms.
Theorem(TheAlternatingSeriesTest)The alternatingseriesu 1 - u 2 +u 3 - u 4 + ... or -u 1 + -u 1 +
u 2 - u 3 +u 4 -... convergeif: 1.u 1 ≥u 2 ≥u 3 ≥ ... ≥uk≥ ... and 2..Take the termsof the seriesand drop their signs.Thenthe theoremtells us that the termsof the seriesmust
be nonincreasingand the limit of the termsis 0 in orderfor the test to work.Hereis an exampleof how to
use The AlternatingSeriesTest.
Example 1
Determinehow if convergesor diverges.
Solution
The series is an alternatingseries.We mustfirst checkthat the termsof the series
are nonincreasing.Notethat in orderforuk≥uuk+ 1, then , or.
So we can checkthat the ratio of the (k+ 1)st term to thekth term is less than or equalto one.
Expandingthe last expression,we get: