HW05-solutions - ybarra (dy3266) HW05 rusin (55735) This...

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ybarra (dy3266) – HW05 – rusin – (55735)1Thisprint-outshouldhave17questions.Multiple-choice questions may continue onthe next column or page – find all choicesbefore answering.00110.0pointsDetermine the interval of convergence ofthe infinite seriessummationdisplayn=1xn(n+ 5)!.1.interval convergence = [-5,5]2.interval convergence = (-5,5)interval convergence = (-5,-3)2.interval convergence = (-5,-3]3.converges only atx= 44.interval convergence = (-∞,)5.interval convergence = (3,5)correct6.interval convergence = [3,5)6.interval convergence = [-1,1]1.interval convergence = (-5,-3)2.interval convergence = (-5,-3]3.converges only atx= 44.interval convergence = (-∞,)5.interval convergence = (3,5)correct6.interval convergence = [3,5)n
00210.0pointsDetermine the interval of convergence ofthe series00310.0pointsDetermine the interval of convergence ofthe infinite series
ybarra (dy3266) – HW05 – rusin – (55735)21.interval conv. =parenleftBig-∞,parenrightBig2.interval conv. =bracketleftBig-115,15parenrightBig3.interval conv. =bracketleftBig15,115parenrightBig4.interval conv. =parenleftBig-115,15bracketrightBigcorrect5.series converges only atx=-16.interval conv. =parenleftBig15,115bracketrightBigExplanation:We apply the Ratio Test to the seriessummationdisplayk=0vextendsinglevextendsinglevextendsinglevextendsingle(-1)k1(k+ 1)6k(5x+ 5)kvextendsinglevextendsinglevextendsinglevextendsingle=summationdisplayk=0|5x+ 5|k(k+ 1)6k.In this casevextendsinglevextendsinglevextendsinglevextendsingleak+1akvextendsinglevextendsinglevextendsinglevextendsingle=|5x+ 5|(k+ 1)6k(k+ 2)6k+1.Thuslimk→ ∞vextendsinglevextendsinglevextendsinglevextendsingleak+1akvextendsinglevextendsinglevextendsinglevextendsingle=16vextendsinglevextendsinglevextendsingle5x+ 5vextendsinglevextendsinglevextendsingle,and so by the Ratio Test the series convergeson the solution set of the inequality|5x+ 5|<6,i.e.,parenleftBig-115,15parenrightBigand diverges on the solution set of|5x+ 5|>6,i.e., onparenleftBig-∞,-115parenrightBiguniondisplayparenleftBig15,parenrightBig.

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Term
Fall
Professor
Chu
Tags
Power Series, Mathematical Series, Ybarra

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