8. InfiniteSeries................................................................................................................................
This chapterintroducesthe studyof sequencesand infiniteseries.In calculus,we are interestedin the be-
haviorof sequencesand series,includingfindingwhethera sequenceapproachesa numberor whetheran
infiniteseriesaddsup to a number. The tests and propertiesin this chapterwill help you describethe behavior
of a sequenceor series.
Sequences
LearningObjectives
- Demonstratean understandingof sequencesand their terms
- Determineif the limit of a sequenceexistsand, if it exists,find the limit
- Applyrules,theorems,and Picard’s methodto computethe limitsof sequences
Sequences(rules,terms,indices)
The alphabet,the namesin a phonebook,the numberedinstructionsof a modelairplanekit, and the
schedulein the local televisionguideare examplesof sequencespeoplemay use. Theseexamplesare all
sets of ordereditems.In mathematics,a sequenceis a list of numbers.You can makefinitesequences,
suchas 2, 4, 6, 8. Thesesequencesend. You can also makeinfinitesequences,suchas 3, 5, 7, 9, ..., which
do not end but continueon as indicatedby the threedots.In this chapterthe wordsequencerefersto an
infinitesequence.
Eachterm in a sequenceis definedby its placeof orderin the list. Considerthe sequence3, 5, 7, 9, .... The
first term is 3 becauseit belongsto place1 of the sequence.The secondterm is 5 becauseit belongsto
the secondplaceof the sequence.Likewise,The third term is 7 becauseit is in the third place.Noticethat
thereis a naturalrelationshipbetweenthe countingnumbers,or the positiveintegers,and the termsof the
sequence.This leadsus to the definitionof asequence.
SequenceAsequenceis a functionfrom the domainof the set of countingnumbers,or positiveinte-
gers,to the rangewhichconsistsof the membersof a sequence.A sequencecan be denotedby {a
n} or bya 1 ,a 2 ,a 3 ,a 4 , ...,an,....The numbersa 1 ,a 2 ,a 3 ,a 4 , ...,an,.... that belongto a sequenceare calledtermsof the sequence.Each
subscriptof 1, 2, 3, ... on the termsa 1 ,a 2 ,a 3 ,a 4 , ...refersto the placeof the termsin the sequence,or the
index.The subscriptsare calledtheindicesof the terms.We assumethatn= 1, 2, 3,..., unlessotherwise
noted.
Insteadof listingthe elementsof a sequence,we can definea sequenceby arule,or formula,in termsof
the indices.
Example 1
The formula is a rule for a sequence.