Geometric series are series of the formin which a and r are fixed real nuimbers and a. *.0Theratiercambepoutive,usi1+#+++1/2,1=2.Geometric Seriesornegatiwg as in1-++--+(-1/3,a-I teh pantial sum ofthe gometricseriesiSa+ a(1) + a(1) +.. +o1-1 m,and the series divenges because lime s, - ±oo, depending on the sign ofa. Ifr = 1,the series diverges because the wth partial sums alternatebetween a and 0Slide 4-31Slide 4-32+aIfir/≤1, thensa+ar+ar+EXAMPLE1The geometric series with a = 1/9 and r 1/3 is+ar ++.+ arr.Muhtiphy J, by1/9+++++*()1-0/) -Iudrsfmm.Motoare5(1 r) = a(1 -uEXAMPLE2The seriesa(l -r)(r→1)2-5--+-+Ii<1.then→0.as 00 (as in Section 10.1) and sg/(1 - r). If| > 1,04thenjr→ oo and the series diverngesisageometricseries wih= Sandr=1/4.Itcovergesto1-,"1+ (/4) = 4,1f]1] < 1, the geometric' series a--comvergestoa/(1 r):aI/<1.-rIf|r] ≥ 1, the series divengesSlide 4- 33Slide 4- 34EXAMPLE3You drop a ballfirom a mcters abovea flat surface. Each time the ballhitsExercisesthesurfaee ifterfllingadistanceh,itreboundsadistance rh, whererispositivebutlessFinding nth Partial Sumsthan 1. Find the total distance the balltravels up and down (Figure 10.9),In Exercises I-6, find a formula for the nth partial sum of each seriesand use it to find the series'sum if the series converges.1.2+1+#+#+++..2++100+100100210033.1-+++-+++(1)-1 +..2814. 1 2 + 4 8 + -+ (1)1 2*1 + ..The total distance isIn Exercises 1518, determine if the geometric series converges or di-inghaBifdel=a+20r+2++2verges.Ifaseries comverges,find its sum15. 1+()+() + () + (图) +If a = 6 m and r = 2/3, foe instance, the distance i+ (2/3)(5/3)16. 1 + (3) + (3) + (3) + (3)* + 4.= 30 -(2/3)-()slide 4. 35Slide 4-38
2016/11/15 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 4 - 31 2. Geometric Series Slide 4 - 32 Slide 4 - 33 Slide 4 - 34 Slide 4 - 35 Slide 4 - 36 Exercises
PxerEisesFind thesum ofthle 'elescoping"seriesEXAMPLE5+1)Find the sum of each series in Exercises 4148SolutionWe look for a pattern in the sequence of partial sums that might lead to a for-8W546mula for Sg, The key observation is the purtial fraction decomposition41.42.台 (4n 3)(4n + 1)台 (2n 1)(2n + 1)+---405_2n +1so43.44. (2n 1)(2# + 1)2台n（n+1)2高品烹(-+)and#-（1)+（送-）+（-）++（Removingparenlsign collapses the sum tcndcingadacfop$-1-本We now seethat S1as k→ oc,The series converges,and its sum is 1:S2+=."Slide 4-37Slide 4-38EXAMPLE6The series+-++++diverges because the partial sums eventually outgrow every preassigned number. Eachtemgreaterhanoesumoftesigeaterth3.Thenth-TermTestforaTHEOREM?Sges, then a,Divergent SeriesThe nth-Term Test for DihergenceiieaSlide 4- 39Slide 4- 40ExercisesEXAMPLE7The following are all examples of divergenit'series.Using the nth-Term TestNr diverges eausen'- 0(a)In Exercises 27-34, use the nth-Term Test for divergence to show thatthe series is divergent, or state that the test is inconclusive.FA+IIdivegsbcause "t!1(b)in on(n+1)5127. 28.高+102(m+2)(n+3)(c)S30.2129.台28+4+3An(d)22 diegs bease im 21 - 0.Slide 4- 41Slide 4- 42
2016/11/15 7 Slide 4 - 37 Slide 4 - 38 Exercises Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 4 - 39 3. The nth-Term Test for a Divergent Series Slide 4 - 40 Slide 4 - 41 Slide 4 - 42 Exercises
THEOREM8I Ea, =Aand b, =Bare comvergeut series, thenLSum Rale:Ea, +b)-Ea, +h,=A+82. Diference Rule:E(a,h)-Ea,-Eh, - A -B3,Constamt Mulriple RaleSo,-a-4(any mumber ),CorollariesEvery nonzero constant multiple ofadivengent seriesdivenges4.Combing Series2. If Za, comerges and Eb, dinerges,them (a, + h) and (a, h) bothdivergeRemember thar E(a+ b) can convege when Za, and b, both divergeCantionForexample,Za,=1+1+1++ and Zb, = (1) + (1) + (1) +--divernge.whereas (a, + b,)=0 + 0 + 0+--coverges to 0.lide 4- 43Slide 4- 44EXAMPLE9Findthe sums ofthefollowing series-()6--Differesce.Ral10.3a1(/2) ~1 (0/6)-2-号-号024-42The Integral TestMunipirRt-(r-(s)cotdc mries wie a.= 1,r = 1/2-8slide 4- 45Slide 4- 48Suppose that Zht a, is an infinite series with a, ≥ 0 for all n. Then each partial sum isgreater than or equal to its predecessor because S+1 S, + ae:SSS≤s≤Since the partial sums form a nondecreasing sequence, the Monotonic Sequence Theorem(Theorem 6, Section 10.1) gives the following result.1.NondecreasingPartial SumsAseriesfoetemmesCarollary af Theorem 6and only if its partial sums are bounded from aboveSlide 4- 47Slide 4.48
2016/11/15 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 4 - 43 4. Combing Series Slide 4 - 44 Corollaries Slide 4 - 45 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 46 10.3 The Integral Test Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 4 - 47 1. Nondecreasing Partial Sums Slide 4 - 48