CK12 Calculus - Single Variable

Computethe sum of a convergentgeometricseries Determineconvergenceor divergenceof seriesusingthe nth-Term Test InfiniteSeries( ...

s 3 =u 1 +u 2 +u 3 The first sum is the first term of the sequence.The secondsum is the sum of the first two terms.The third ter ...

Doesthe infiniteseries1 + 0.1 + 0.01 + 0.001+ ... convergeor diverge? Solution To makeour workeasier, writethe infiniteseries1 + ...

Solveforsn by multiplyingboth sidesof the last equationby : Now we find the limit of both sides: The sum of the infiniteseriesis ...

The convergenceor divergenceof a geometricseriesdependsonr. TheoremSupposethat the geometricseries has ratior. 1. The geometrics ...

The originalproblem, , doesnot havethe leading term of 3. This doesnot affect the convergencebut will affect the sum of the seri ...

. Then and. OtherDivergentSeries(nth-Term Test) Determiningconvergenceby usingthe limit of the sequenceof partialsumsis not alwa ...

Usingthenth-Term Test,. Sincethe limit is 0, we cannotmakea conclusionaboutconvergence or divergence. Rulesfor ConvergentSeries, ...

Example 11 Find the sum of. Solution By the RulesTheorem,. The series is a geometricserieswitha = 5 and. The sum of is. Then. Ad ...

thoughfor reindexing. Example 13 is a convergentgeometricseries.It can be reindexedby changingthe startingpositionofiand the pow ...

Suppose is a convergentseriesand is a divergentseries.Explainwhy and both diverge. Give an exampleof a geometricserieswhosesu ...

Then The seriesconverges. 4.a= 3,r= 2 The series is a geometricserieswitha= 1 and and so, the geometric seriesconverges.The su ...

Then. Since is divergent,then or. Since is convergent, is finite.Then is the sum of somethingfiniteand something infinite.Thus ...

divergence geometricseries ratio of geometricseries nth-Term Test reindexing SeriesWithoutNegativeTerms LearningObjectives Demo ...

Write the first few terms:. The sequenceis nondecreasing.To determineconver- gence,we see if we can find a constantBsuchthat. If ...

Example 3 The geometricseries has all non-negativeterms.The sequenceof partialsums lookslike this: Intuitively, we can see that ...

Rewrite as to get. The valueofpis. This is less than 1, whichtells us that the seriesdiverges. ComparisonTest Now that we havest ...

The series is similarto. Usingthe ComparisonTest, for allk. The series divergessinceit is ap-serieswith. By the ComparisonTest, ...

Then. Thus,. Sincethe integralis finite,the series convergesby the IntegralTest. LimitComparisonTest, SimplifiedLimitComparisonT ...

ratio of the termsof the two seriesis positive: . Usingthe Limit ComparisonTest, because divergesand the limit of the ratio is p ...