divergence
geometricseries
ratio of geometricseries
nth-Term Test
reindexing
SeriesWithoutNegativeTerms
LearningObjectives
- Demonstratean understandingof nondecreasingsequences
- Recognizeharmonicseries,geometricseries,andp-seriesand determineconvergenceor divergence
- Applythe ComparisonTest, the IntegralTest, and the LimitComparisonTest
NondecreasingSequences
In orderto extendour studyon infiniteseries,we mustfirst take a look at a specialtype of sequence.
NondecreasingSequenceAnondecreasingsequence{Sn} is a sequenceof termsthat do not decrease:. Eachterm is greaterthan or equalto the previousterm.
Example 1 5, 10, 15, 20, ... is a nondecreasingsequence.Eachterm is greaterthan the previousterm: 5
< 10 < 15 < 20 < ....
10,000,1000,100, ... is not a nondecreasingsequence.Eachterm is less than the previousterm:10,000,
1000,> 100 ....
3, 3, 4, 4, 5, 5, ... is a nondecreasingsequence.Eachterm is less than or equalto the previousterm:3 ≤ 3
≤ 4 ≤ 4 ≤ 5 ≤ 5 ≤ ....
A discussionaboutsequenceswouldnot be completewithouttalkingaboutlimits.It turnsout that certain
nondecreasingsequencesare convergent.
TheoremLet {Sn} be a nondecreasingsequence:S 1 ≤S 2 ≤S 3 ≤ ... ≤Sn≤ .... 1. If there
is a constantBsuchthatSn≤Bfor alln, then existsand whereL≤B. 2. If the
constantB doesnot exist,then.The theoremsaysthat a bounded,convergent,nondecreasingsequencehas a limit that is less than or
equalto the bound.If we cannotfind a bound,the sequencediverges.
Example 2 Determineif the sequence convergesor diverges.If it converges,find its limit.
Solution