Part I: Covers Sequence through Series Comparison Tests

Transcription

1 Part I: Covers Sequece through Series Compariso Tests. Give a example of each of the followig: (a) A geometric sequece: (b) A alteratig sequece: (c) A sequece that is bouded, but ot coverget: (d) A sequece that is mootoic, but ot coverget: (e) A sequece that is ot bouded ad or mootoic:

2 2. For each of the sequeces below, determie if they are coverget. If they are, fid their limit. Remember a few thigs: i) If the sequece looks like somethig we could have take the limit of i Calculus, the you ca fid the limit the same way you would have i Calculus. ii) Factorials grow faster tha polyomials ad expoetial fuctios, but ot as fast as thigs that look like. iii) If the sequeces is recursive, you usually determie covergece by tryig to show the sequece is bouded ad mootoic. If it is, you ca the fid its limit L from the recursio formula, by replacig each istace of the sequece with the limit L ad the solvig for L. (a) a = (b) b = 2 3 (c) d = l(3 + ) l() 3. What is ? (it is bouded above) Page 2

3 4. For each of the series below, compute the first 5 partial sums S through S 5. Do you thik these series coverge or diverge? (a) 2 + (b), 000, 000 (c) Which of the followig would chage whether or ot a give series coverges? Startig the series at = 5 istead of =. Deletig a millio terms from the series. Addig a millio terms to the series. Page 3

4 6. Does the sum of two coverget series always coverge? Does the sum of two diverget series always diverge? What happes if you add a coverget series ad a diverget series together? 7. For each of the followig series, determie if they coverge or diverge. If they coverge, use the geometric series formula ad/or the telescopig sum techiques to compute the sum exactly. (a) =2 (b) (c) 0 arcta Page 4

5 (d) (e) Page 5

6 8. Oe applicatio of summig geometric series is to covert repeatig decimals ito their fractioal couterparts. For istace, the decimal 0.7 is really just the ifiite series Which we ca rewrite as 7 0 Summig this series, we obtai that the above repeatig decimal is just 7 9. Use this to fid the fractioal represetatios of the repeatig decimals below: (a) 0.9 (b) 0.6 Page 6

7 9. The Itegral Test: The itegral test says that if f() is a sequece that satisfies a couple of properties, the f() coverges. What are these properties? =0 If g(x) is cotiuous ad decreasig: 0. If 0 g(x) f(x) ad k g(x) dx coverges, the f() coverges. =k (T/F). If 0 f(x) g(x) ad k g(x) dx coverges, the f() coverges. =k (T/F) 2. Sice x dx =, does this mea that 2 we fid the sum of =? Why or why ot? If ot, ca 2 usig the techiques we leared i this class? 2 Page 7

8 3. Use the Itegral Test to determie whether each of these series coverge or diverge: (a) e + 2 (b) Use the result from above, determie if e / + 2 coverges. Page 8

9 4. The p-test: Sums of the form, or commoly appear i applicatios. These ca be hadled by the Itegral Test; however, whether or ot they p (l()) p coverge due the Itegral Test depeds o the value of p. Rather tha do the Itegral Test every sigle time, we just remember the values of p for which these kids of series coverge. Fill i the blaks below: If p, the If p >, the If If, the, the p p (l ) p diverges. (l ) p coverges. 5. Use the p-test to quickly determie if the followig itegrals coverge or diverge: (a) (b) (c) (d) (e) =8 =8 =8 =8 5 /2 l (l ) 3/2 2 (l ) (l ) Page 9

10 6. Workig With Iequalities: A Primer Whe usig the Direct Compariso Test to show that a series a, where a > 0, coverges or diverges, you are lookig to fid aother sequece b, where b > 0, such that: a b ad b coverges. This shows that a coverges. or that: b a ad b diverges. This shows that a diverges. But how do you show that a b or that b a? Usually, you will have a guess for what b should be. The, startig with a, you will maipulate the formula for a i ways that will make it either larger or smaller util you reach your target. The followig page will explai some ways to do this. How to make somethig bigger Add somethig positive to it (Ex: a a + ) Remove somethig egative from it. (Ex: a a ) Replace a term with somethig bigger. (Ex: = 2 2 ) If you have a fractio, make the umerator larger. If you have a fractio, make the deomiator smaller. If you have some other icreasig fuctio, such as a square root, you ca make the stuff iside of it larger. (Ex: < + ). How to make somethig smaller Add somethig egative to it (Ex: a a ) Remove somethig positive from it. (Ex: a + a ) Replace a term with somethig smaller. (Ex: 2 + si() 2 ) If you have a fractio, make the umerator smaller. If you have a fractio, make the deomiator larger. If you have some other icreasig fuctio, such as a square root, you ca make the stuff iside of it smaller. (Ex: + > ). Page 0

11 Strategy for Direct Compariso Idetify the terms that grow the fastest i both the umerator ad deomiator. Their ratio will usually form the sequece b. Determie if b coverges or diverges. If b coverges, the use the make somethig bigger strategies above to remove terms or replace terms of a util you get b. If b diverges, the use the make somethig smaller strategies above to remove terms or replace terms of a util you get b. 2 + Example: Determie the covergece of The largest term i the umerator is 2, ad the largest i the deomiator is 3. 2 So this should behave like = 3 = 3, ad coverges. So we wat to make a = larger util we get to b =. 2 Usig our rules above, we have that (makig umerator larger) (makig deomiator smaller) (makig deomiator eve smaller) a = (simplifyig) = = = 2b The 2 i frot of the b does t matter, as 2b coverges if b does. Thus, sice 2b coverges, so does a. The poit of the legthy discussio above is to show you that i order to use the direct compariso test, it is very tedious to fid the correct size for b. I hope it s eough to coveiece you that, uless you NEED to use the direct compariso test whe it s easy to fid a upper ad lower bouds for the a, it s much easier to use the limit compariso where you do ot have to deal with the size. Page

12 7. Determie the covergece of each of the followig series usig Direct Compariso: (a) (b) =2 2 l (c) 0 cos 2 (d) ( ) + 4 l Page 2

13 Limit Compariso: Sometimes, though, you will kow what you wat to compare your series to, but you caot get Direct Compariso to work. (Or work easily, i ay case). I this case, you should try Limit Compariso istead of Direct Compariso. The steps are pretty similar: Like you would for Direct Compariso, idetify the terms that grow the fastest i both the umerator ad deomiator. Their ratio will usually form the sequece b. (We may phrase this as figurig out what the series behaves like ). Fid lim. If this limit exists, is fiite, ad is larger tha 0, the the two series b a ad b both coverge or both diverge. Essetially, this limit calculatio shows that the two sequeces decay at the same rate, ad so their series will grow at the same rate. a Limit Compariso is ofte better tha Direct Compariso for series ivolvig complicated ratioal fuctios, as well as series where you do t have a good guess for somethig to compare directly with. a 8. If lim = 0 ad b coverges, the a coverges. Explai why this is the case. b a 9. Similarly, if lim = ad a coverges, the b coverges. Explai why this b is the case. Page 3

14 20. For each of the positive-term series a below, we ca determie covergece by comparig to aother series b. Choose the sequece b to compare to, say whether b coverges or diverges, ad idicate whether you would use the Direct Compariso Test or the Limit Compariso Test to carry out the justificatio. Also give the iequality you will use for Direct Compariso Test, or the value of the limit, for the Limit Compariso Test. (a) (e) l 3 (b) si ( 3 ) (f) ( ) si 3 (c) (g) =2 arcta 3 (d) l (h) si Page 4

15 (a) si 3 (b) = si (c) 5 2 (d) ( ) ta 3 Page 5

16 Exam 2 Review. Determie whether the sequece coverges or diverges. If it coverges, fid the limit a. a = (2 )! (2+)! b. a = cos2 2 c. a = ( )+ +2 d. a =! 2 e. a = arcta(l ) f. a = g. a = l(4) l(4 ) h. a = (l ) i. a = ( + 3 )2 2. Fid the limit L of the sequece, or say DIV. (the recursive sequeces (a) ad (d) are coverget). { a. a =, a + = 3 a b. 2, 2 + 4, } 8,... c. { 3, , } 3 5,... d. { 5, 5 5, 5 5 5,... } 3. i. What happes to the series ii. What happes to the series iii. Suppose S N = lim a, S N, N= a if lim a = 0? a if lim a 0? N a ad that S N = 5 N. What ca be said about 2N! 6 (a + )? Evaluate a i. iv Suppose S N = arcta, the lim a = 0. TRUE/FALSE? i=3

17 4. Suppose you kow that a <, b diverges. Which statemet below is TRUE? i If a < c, the c diverges. ii If c < b, the c coverges. iii If c > b, the c diverges. iv If c < a, the c diverges ca be show coverget usig DCT by comparig with 42 5 TRUE or FALSE? Determie the value of k for which the series your aswer i iterval otatio. 7. Does the series coverge? If so, fid the sum. a b. = c. ( ) k +24 will coverge. Write d. 3 (+3) 8. Which of the followig statemets are true? I. II. III. (l ) 347 =5 ( ) =5(l ) 4 (l ) 2 =5 coverges by the Direct Compariso Test. coverges by the Alteratig Series Test. coverges by the Direct Compariso Test. 2

18 IV. 2 =5 3 l coverges by the Direct Compariso Test. 9. Use Root Test: Which of the followig series coverge? Fid the correct limit of the test. ( a. 0 ) ( b. 7 ) c. 9+0 (l ) d. 2! =2 e f. + ( ) 5 g. =4 + h. 2 (l ) 3 =2 i. ( + ) 2 9. Use Ratio Test: Which of the followig series coverge? Fid the correct limit of the test. 3 5 (2 + ) (2)! a. b (3 + 2) (!) 2 3

19 0. Alteratig series test. Determie if the series coverges absolutely, coditioally, or diverges. a. ( ) l b. ( ) + (l ) c. =2 =2 ( ) + (l ). Accordig to the alteratig series error estimatio, what is the least upper estimates to the error by usig the first 4 terms to approximate the sum? ( R 4 ) a. b. ( ) + ( ) +! 2. Determie whether the series is coverget or diverget usig Direct Compariso or Limit Compariso. a. +cos e b c. ( ta ) d. ta l ( ) 2 e. ( ) + 2 f. l 5 g. l 2 3. Does the series coverge? what test(s) do you use? a. ta ( ) b. cos ( ) c. l =2 d. ( ) cos ( 2 ) e. si ( ) f. si 3( ) 4

20 4. Which of the followig series are diverget by the Test for Divergece? a. / b c. 2 e d. ta ( ) 5. Determie whether each series is absolutely coverget, coditioally coverget or diverget. Be clear i your argumet ad ote what test(s) you use. a. b. =5 =5 ( ) arcta 2 ( ) + 2/3 l 0 5