Sincekis positiveand all the sum of the numeratorare part of the denominator’s sum,the numeratoris less
than the denominatorand so,. Thus,uk≥uk+ 1for allk. By the AlternatingSeriesTest, the
series converges.
Keepin mindthat both conditionshaveto be satisfiedfor the test to proveconvergence.However, if the
limit conditionis not satisfied,the infiniteseriesdiverges.
AlternatingSeriesRemainder
We find the sequenceof partialsumsfor an alternatingseries.A partialsum can be usedto approximate
the sum of the series.If the alternatingseriesconverges,we can actuallyfind a boundon the difference
betweenthe partialsum and the actualsum.This difference,or remainder, is calledtheerror.
Theorem(AlternatingSeriesRemainder)Supposean alternatingseriessatisfiesthe conditionsof
the AlternatingSeriesTest and has the sum. Let be thenth partialsum of the series.Then
.The mainidea of the theoremis that the remainder cannotget largerthan the term in
the series,. This is the term whoseindexis one morethan the indexof the partialsum usedin the
difference.
Example 2
Computes 3 for the series and determinethe boundon the remainder.
Solution
First we computethe third partialsum to approximatethe sumSof the series:
The theoremtells us to use the next term in the series,u 4 , to calculatethe boundon the differenceor re-
mainder. Rememberthat the part (-1)k+1just givesthe sign of the term and, so we just use the part
to calculateu 4.