CK12 Calculus - Single Variable

Considerthe sequence{n+ 1} in Figure4. Asngets largerand goesto infinity, the termsofan=n+ 1

becomelargerand larger. The sequence{n+ 1} doesnot havea limit. We write
Convergenceand Divergence
We say that a sequence{an}convergesto a limitLif sequencehas a finitelimitL. The sequencehas
convergence.We describethe sequenceasconvergent.Likewise,a sequence{an} divergesto a limit
Lif sequencedoesnot havea finitelimit. The sequencehasdivergenceand we describethe sequenceas
divergent.
Example 7
The sequence{ln (n)} growswithoutboundasnapproachesinfinity. Notethat the relatedfunctiony= ln(x)
growswithoutbound.The sequence is divergentbecauseit does not havea finite limit.We write
.
Example 8

The sequence convergesto the limitL= 4 and henceis convergent.If you graphthe function

forn= 1, 2, 3,..., you will see that the graphapproaches4 asngets larger. Algebraically, asn

goes to infinity, the term gets smaller and tends to 0 while 4 stays constant. We write

.
Example 9
Doesthe sequencesnwith terms1, –1, 1, –1, 1, –1, .... havea limit?

Solution
This sequenceoscillates,or goesbackand forth,betweenthe values1 and –1. The sequencedoesnot get
closerto 1 or –1 asngets larger. We say that the sequencedoesnot havea limit,or doesnot
exist.
Note:Eachsequence’s limit falls underonly one of the four possiblecases:

1. A limit existsand the limit isL:.


2. Thereis no limit: doesnot exist.


3. The limit growswithoutboundin the positivedirectionand is divergent:.


4. The limit growswithoutboundin the negativedirectionand is divergent:
   If a sequencehas a finitelimit, then it only has one valuefor that limit.