CK12 Calculus - Single Variable

Determineif convergesor diverges.If it converges,find its limit. Determineif convergesor diverges.If it converges,find its sum. ...

is a geometric series with a = 1 and. The sum is . divergesby Comparisontest (using , a divergentp-series). convergesby the L ...

intermediate intermediate challenging Keywords nondecreasingsequence harmonicseries geometricseries 4.p-series ComparisonTest ...

. The odd-indexedtermsof this serieshavethe negativesign. Thealternatingp-seriesis anothertype of alternatingseries.An exampleco ...

Sincekis positiveand all the sum of the numeratorare part of the denominator’s sum,the numeratoris less than the denominatorand ...

Thus. Then. This tells us that the absolute valueof the erroror remainderis less than 0.13. Absoluteand ConditionalConvergence T ...

Determineif the series convergesabsolutely. Solution The seriesmadeup of the absolutevaluesof the termsis This seriesbehaveslike ...

This tells us that rearrangementdoesnot affect absoluteconvergence. ReviewQuestions Determineif the seriesconvergesor diverges. ...

10. 11. 12. Answers convergesby the AlternatingSeriesTest. convergesby the AlternatingSeriesTest. 19 199 9,999 ...

challenging challenging challenging challenging intermediate challenging intermediate Keywords alternatingseries alternatingha ...

is absolutelyconvergent,thenit is convergent. The proofis quitestraightforwardand is left as an exercise.The converseof Theorem5 ...

Solution.Let. Then asn→ ∞. So by the RatioTest, the seriesis absolutelyconvergentfor any constantA. Indeed,the sum is eAwhichis ...

(B) If or , then the seriesis absolutelydivergent. (C) If , then the test is inconclusive. The proofis similarto that of the Rat ...

Hint: and existsand equalse. Summaryof Proceduresfor DeterminingConvergence We haveseenvarioustest for convergenceof in action.T ...

and applyingLimit Comparisontest with . The Root Test is inconclusiveon , but the simplerTest for Divergenceconfirmsits divergen ...

For whatvaluesofcis the series convergent? PowerSeries PowerSeriesand Convergence Definition(PowerSeries) APowerSeriesis a ser ...

Let Then is the powerseriesfor on. Let Then is the power series for on (1,3). So and are the powerseriesfor the samefunction but ...

Example 1 Find the radiusof convergenceand intervalof convergenceof the series. Solution.Let. Then as. So the seriesis absolutel ...

e. Term-by-Term Differentiationof PowerSeries The goal of the next 3 sectionsis to find powerseriesrepresentationsof certainclas ...

Exercise Find a powerseriesand the radiusof convergencefor the followingfunctions: Term-by-Term Integrationof PowerSeri ...