10.5 Test Info. Test may change slightly.

Transcription

1 0.5 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 Remider (ltertig) o Types of questios Fid remider (ext term up) Approximte usig prtil sums (prtil sum ext term < S < prtil sum + 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)

2 SUMMARY OF TESTS FOR SERIES TEST SERIES CONVERGES DIVERGES COMMENT th Term Test for Divergece (60) 0 This test CANT be used to show covergece Telescopig (607) ( b b ) b L Sum S b L Geometric Series (608) r r < r > 0 S Sum r 0.5. Itegrl Test f is positive, cotiuous, & decresig for x. (67) f ( ) 0 f ( x) dx coverges f ( x) dx diverges Remider 0 RN f ( x) dx N p-series (69) p p > p < Direct Compriso (, b 0) (64) 0 b AND b coverges 0 b AND b diverges If the lrger series coverges, the smller series must coverge. If the smller series diverges, the lrger series must diverge Limit Compriso (, b 0) (66) b AND L b b AND L or L is fiite d positive. b coverges b diverges

3 SUMMARY OF TESTS FOR SERIES TEST SERIES CONVERGES DIVERGES COMMENT Altertig Series Test (6) ( ) 0 AND 0 Remider R N The Rtio Test (69) Test is icoclusive if The Root Test (64) Test is icoclusive if Absolute Covergece Theorem: If the series coverges, the the series lso coverges. Defiitios of Absolute d Coditiol Covergece:. is bsolutely coverget if coverges.. is coditiolly coverget if coverges but diverges.

4 Series Covergece/Divergece Flow Chrt TEST FOR DIVERGENCE Does = 0? Diverges p-series Does = / p,? Is p >? Coverges Diverges GEOMETRIC SERIES Does = r,? Is r <? = = r Diverges ALTERNATING SERIES Does = ( ) b or = ( ) b, b 0? Is b + b & b = 0? Coverges TELESCOPING SERIES Do subsequet terms ccel out previous terms i the sum? My hve to use prtil frctios, properties of logrithms, et 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: COMPARISON TEST Pick {b }. Does b coverge? Is 0 b? Is 0 b? Coverges Diverges LIMIT COMPARISON TEST Pick {b }. Does c fiite &, b > 0? b = c > 0 Does b coverge? = Coverges Diverges INTEGRAL 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

5 Problems -8 from Stewrt s Clculus, pge 784. = + 4. si() = 7. k= k l(k) (k + ) = = + + ( ) + = ( ) + = = = k= ( ) + 8 l() k k! (k + )! k e k k= e = = ( ) + l() ( ) + 5 = =0 =! 5 8 ( + ) + + ( ) / = = ( ) ( ) l() = k= k k ( ) = = t(/) = = e = cos(/) + 4! = = e / t () j ( ) j j + 5 j= k= = = = = = 5 k k + 4 k () si(/) + cos () ( ) + (l()) l() ( ) =. =! 6. = ( ) =

6 Review Questios 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 ONCE. Fid the sum, where idicted. )! 58 0 ) 5 ) 4) 5 5

7 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, ONLY!]

8 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: 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 cos !

9 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 r d. 0r e. r 8. The sum of the geometric series... is b. d. e Which of the followig sttemets bout series is true?. If u 0, the u coverges. b. If u 0,the u diverges. u { r } 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. d. 4 e. oe of these. Which of the followig series diverges? b d.... e Which of the followig series diverges?. b. d. e.! l l

10 . For which of the followig series does the Rtio Test Fil?.! b l l l 4...! d. 4 e. 4. The sum of the series is equl to. 0 b. d. e. oe of these 5. Whe is pproximted by the sum of its first 00 terms, the error is closest to b d. 0.0 e Which of the followig series diverge? I. k II. III. k k k k 6 7 k k. Noe b. II oly III oly d. I d III e. II d III If s 00, to wht 5 4 umber does the sequece s coverge?. 5 b d. 4 e. The sequece does ot coverge 8. For wht iteger k, k, will both k k d 4. 6 b. 5 4 d. e. coverge?

11 Altertig Series Remider If coverget ltertig series stisfies the coditio, the the bsolute vlue of the remider RN ivolved i pproximtig the sum S by SN is less th or equl to the first eglected term. S S N R N N 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 N RN N to determie the umber of terms required to pproximte the sum of the coverget series with error of less th Use TI-8 to pproximte the sum of the series with error of less th ( )! e