9. 9.6 Test Ifo Test my chge slightly. Short swer (0 questios 6 poits ech) o Must choose your ow test o Tests my oly be used oce o Tests/types you re resposible for: Geometric (kow sum) Telescopig (kow sum) p-series th term Itegrl test Direct compriso test Limit compriso test Altertig series test Rtio test Root test Short swer ( questios 6 poits ech) o Remiders (ltertig d itegrl tests) o Types of questios Fid remider Itegrl from (stop) to ifiity ext term up for ltertig Approximte usig certi mout of terms Sum of terms < S < Sum of terms + itegrl Sum of terms ext term < S < Sum of terms + ext term Fid out how my terms it would tke to get to certi remider Multiple Choice (6 questios poits ech) o AP questios o questio bout sequece covergece Test will be out of 00 poits (90 poits o questios, 0 poits for showig up) Extr prctice problems (textbook) Sectio Pge umber Questios 9. Sequeces pg. 60 #, 7, 9, 4, 47, 49, 5, 57, 6, 65, 85, 89, 9 9. Series pg. 6 #7, 9,, 5, 7,, 7, 7, 45, 6, 6, 67, 69 9. Itegrl Test d p-series pg. 60 #, 7,, 5, 8, 9,,,, 6, 6, 7 9.4 Compriso Tests pg. 68 #7, 9,,, 7, 9,, 9.5 Altertig Series pg. 66 #, 7, 9,, 5, 7, 9,, 5, 7, 4, 4, 47, 49, 5, 5, 57, 59 9.6 Rtio & Root Test pg. 656 #5,, 7, 9, 5, 9, 5, 7, 4, 4, 47, 5, 55, 57, 6, 6, 67, 77
SUMMARY OF TESTS FOR SERIES TEST SERIES COVERGES DIVERGES COMMET th Term Test for Divergece (60) 0 This test CAT be used to show covergece. 9. Telescopig (607) ( b b ) b L Sum S b L Geometric Series (608) r r < r > 0 S Sum r 9. Itegrl Test f is positive, cotiuous, & decresig for x. (67) f ( ) 0 f ( x) dx coverges f ( x) dx diverges Remider 0 R f ( x) dx p-series (69) p p > p < Direct Compriso (, b 0) (64) 0 b AD b coverges 0 b AD b diverges If the lrger series coverges, the smller series must coverge. If the smller series diverges, the lrger series must diverge. 9.4 Limit Compriso (, b 0) (66) b AD L b b AD L or L is fiite d positive. b coverges b diverges
SUMMARY OF TESTS FOR SERIES TEST SERIES COVERGES DIVERGES COMMET 9.5 Altertig Series Test (6) ( ) 0 AD 0 Remider R The Rtio Test (69) Test is icoclusive if 9.6 The Root Test (64) Test is icoclusive if Absolute Covergece Theorem: (64) If the series coverges, the the series lso coverges. Defiitios of Absolute d Coditiol Covergece: (64). is bsolutely coverget if coverges.. is coditiolly coverget if coverges but diverges.
Series Covergece/Divergece Flow Chrt TEST FOR DIVERGECE Does = 0? Diverges p-series Does = / p,? Is p >? Coverges Diverges GEOMETRIC SERIES Does = r,? Is r <? = = r Diverges ALTERATIG SERIES Does = ( ) b or = ( ) b, b 0? Is b + b & b = 0? Coverges TELESCOPIG SERIES Do subsequet terms ccel out previous terms i the sum? My hve to use prtil frctios, properties of logrithms, etc. to put ito pproprite form. Does s = s s fiite? = s Diverges TAYLOR SERIES Does = f() ()! (x )? Is x i itervl of covergece? =0 = f(x) Diverges Try oe or more of the followig tests: COMPARISO TEST Pick {b }. Does b coverge? Is 0 b? Is 0 b? Coverges Diverges LIMIT COMPARISO TEST Pick {b }. Does c fiite &, b > 0? b = c > 0 Does b coverge? = Coverges Diverges ITEGRAL TEST Does = f(), f(x) is cotiuous, positive & decresig o [, )? Does f(x)dx coverge? = Coverges Diverges RATIO TEST Is + /? Is + <? Abs. Cov. Diverges ROOT TEST Is? Is <? Abs. Cov. Diverges
Problems -8 from Stewrt s Clculus, pge 784. = + 4. si() = 7. k= k l(k) (k + ).. 4. 5. 6. 7. 8. 9. 0... = = + + ( ) + = ( ) + = = = k= ( ) + 8 l() k k! (k + )! k e k k= e = = ( ) + l() ( ) + 5 = 5. 6. 7. 8. 9. 0.... 4. 5. =0 =! 5 8 ( + ) + + ( ) / = = ( ) ( ) l() = k= k + 5 5 k ( ) = = + + 5 t(/) = = e = cos(/) + 4! 8. 9. 0.... 4. 5. 6. 7. = = e / t () j ( ) j j + 5 j= k= = = = = = 5 k k + 4 k () si(/) + cos () ( ) + (l()) l() ( ) =. =! 6. = + 5 8. ( ) =
Review 9. 9.6 Show ll of the work tht justifies your swers. Prove covergece or divergece of ech series. Where pproprite, justify bsolute or coditiol covergece. You must cite theorem d clerly show its coditios to receive full credit For #-7: you my use test oly OCE. Fid the sum, where idicted. )! 58 0 ) 5 ) 4) 5 5
5) e 6) Fid the sum. 7) Fid the sum. 8) Approximte the sum of usig the first 6 terms. Wht is the error i your pproximtio? [You my use clcultor for this problem, OLY!]
Extr Review Questios. Determie if the followig sequece coverge or diverges. If the sequece coverges, fid its it. e,,,,.... Fid the sum: 4 6. Use sigm ottio to write the sum: 9 7 8 6 4 Test for covergece or divergece. Idetify the test used. If possible, give the sum of the series. th term test for Divergece Geometric Series Test p-series Test Telescopig Series Test Itegrl Test Direct Compriso Test Limit Compriso Test Altertig Series Test Rtio Test Root Test 4. 0 7 0.... 8 4 5.. 5 6 78 6. 5. cos 7. 5 5. 4 8.! 4. 5 8 6 9 4 5 9.... 5 6 7 8 9 5. 5
6. Determie if the series coverges bsolutely, coverges coditiolly, or diverges. 5 4 Prctice Multiple Choice: 7. The sequece coverges if d oly if. r b. r c. r d. 0r e. r 8. The sum of the geometric series... is 4 8 4. 5 b. c. d. e. 4 9. Which of the followig sttemets bout series is true?. If u 0, the u coverges. b. If u 0,the u diverges. u { r } c. If u diverges, the u 0, d. coverges if d oly if u 0. e. oe of these 0. Which of the followig series diverges?. b. c. d. 4 e. oe of these. Which of the followig series diverges?.... 9 7 b.... 4 c.... 4 d.... e.... 4 4 5. Which of the followig series diverges?. b. c. d. e.! l l
. For which of the followig series does the Rtio Test Fil?.! b.... 4 l l l 4...! c. d. 4 e. 4. The sum of the series is equl to. 0 b. c. d. e. oe of these 5. Whe is pproximted by the sum of its first 00 terms, the error is closest to. 0.00 b. 0.00 c. 0.005 d. 0.0 e. 0.0 6. Which of the followig series diverge? I. k II. III. k k k k 6 7 k k. oe b. II oly c. III oly d. I d III e. II d III 00 5 5 7. If s 00, to wht 5 4 umber does the sequece s coverge?. 5 b. 5 c. 4 00 5 d. 4 e. The sequece does ot coverge 8. For wht iteger k, k, will both k k d 4. 6 b. 5 c. 4 d. e. coverge?
Itegrl Series Remider Let f be positive, cotiuous, d decresig fuctio for x, such tht f (). If the series coverges to S, the the Remider R S S is bouded by 0 R f ( x) dx. Use this fct to pproximte the sum of the coverget series usig the idicted umber of terms. Iclude estimte of the mximum error for your pproximtio. (te terms) Altertig Series Remider If coverget ltertig series stisfies the coditio, the the bsolute vlue of the remider R ivolved i pproximtig the sum S by S is less th or equl to the first eglected term. S S R The error i ltertig series prtil sum is less th the bsolute vlue of the ext term. Use the Altertig Series Remider Theorem S S R to determie the umber of terms required to pproximte the sum of the coverget series with error of less th 0.00. Use TI-8 to pproximte the sum of the series with error of less th 0.00. 0 ( )! e