MA 4 Exam Fall 06 Exam Name: Sectio ad/or TA: Do ot remove this aswer page you will retur the whole exam. You will be allowed two hours to complete this test. No books or otes may be used. You may use a graphig calculator durig the exam, but NO calculator with a Computer Algebra System (CAS) or a QWERTY keyboard is permitted. Absolutely o cell phoe use durig the exam is allowed. The exam cosists of multiple choice questios ad 6 free respose questios. Record your aswers to the multiple choice questios o this page by fillig i the circle correspodig to the correct aswer. Show all work to receive full credit o the free respose problems. The wise studet will show work for the multiple choice problems as well. Multiple Choice Questios A B C D E A B C D E A B C D E 4 A B C D E 5 A B C D E 6 A B C D E 7 A B C D E 8 A B C D E 9 A B C D E 0 A B C D E A B C D E A B C D E SCORE Multiple Total Choice 4 5 6 7 8 Score 6 5 5 4 6 00
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MA 4 Exam Fall 06 Multiple Choice Questios e. Cosider the series. If the ratio test is applied to the series, which of the followig iequalities results, implyig that the series coverges?! e A. lim! <! B. lim e < + C. lim e e D. lim E. lim < + < e ( + )! <. The iterval of covergece of the power series A. [0] ( B., ) C. (, ] D. (, ) E. (, +) ( x ) is =0 Page of
MA 4 Exam Fall 06. The sum of the ifiite geometric series + 5 + 4 5 + 8 5 + 6 65 + is A. 5 B. C. 5 D. E. 5 4. Which of the followig sequeces coverge? { } 5 I. { e } II. { e } III. + e A. I oly B. II oly C. I ad II oly D. I ad III oly E. I, II, ad III Page 4 of
MA 4 Exam Fall 06 M 5. If lim M A. B. C. D. E. dx xp coverges, the which of the followig must be true? p coverges. p diverges. coverges. p coverges. p diverges. p+ 6. A series a is coverget if ad oly if a A. the limit lim + a is greater tha. B. its sequece of terms {a } coverges to 0. C. its sequece of partial sums {S } coverges to some real umber. D. its sequece of terms {a } is alteratig. E. its sequece of partial sums {S } is bouded. Page 5 of
MA 4 Exam Fall 06 7. Which of the followig statemets is true? (There is oly oe.) A. If 0 b a ad b coverges the a coverges. B. If lim a = 0 the the series a is coverget. C. The series si is coverget. D. If a is coverget for a > 0 the ( ) a is also coverget. E. The ratio test ca be used to show that coverges. 0 8. Let S N be the N-th partial sum of the series Thus, S =, S =. Compute S 50 S 49. A. 99. B. 50 C. D. 960 E. 0 ( ). Page 6 of
MA 4 Exam Fall 06 9. Cosider the series leads to the followig coclusio. 4 A. The test is icoclusive. 4. Applyig the compariso test with the series + 6 4 B. The series coverges absolutely. C. The series coverges coditioally. D. The series diverges. E. The test caot be applied to a = 4 + 6 4 ad b = 4. 0. The radius of covergece for the series A. B. /0 C. 0 D. /0 E. x =0 0 is Page 7 of
MA 4 Exam Fall 06. The series + =0 4 + A. coverges by the Ratio Test. B. diverges by the Itegral Test. C. coverges by the Limit Compariso Test with the series D. diverges by the Limit Compariso Test with the series E. diverges because it does ot alterate i sig... cos(π). The series is A. coverges absolutely. B. coverges coditioally. C. diverges. D. evetually oscillates betwee ad, but ever coverges. E. oe of the above. Page 8 of
MA 4 Exam Fall 06 Free Respose Questios. Fid the first four (4) terms of each of the followig sequeces. (a) (6 poits) a = ( + )! Solutio:,,!, 4! (b) (6 poits) a = ad a + = a Solutio:,,, 5, 5 4. Determie if the sequece is coverget or diverget. If coverget give its limit. (a) (4 poits) a = + Solutio: The sequece coverges. lim + = lim + =. (b) (4 poits) a = e Solutio: The sequece coverges. lim e = 0. (c) (4 poits) a = Solutio: The sequece diverges. lim = lim ( ) = +. Page 9 of
MA 4 Exam Fall 06 5. Determie the covergece or divergece of each of the followig series. State clearly what test you used ad show your work. (a) (5 poits) Solutio: This series diverges by the p-series test with p = /. (b) (5 poits) si () Solutio: si () si () The latter series coverges by the p-series test with p =, so the give series coverges by the Compariso Test with the series. (c) (5 poits) Solutio: Use the Ratio Test. lim a + a = lim = lim ( + ) + ( + ) = < Sice the limit is less tha, the series coverges by the Ratio Test. Page 0 of
MA 4 Exam Fall 06 6. (5 poits) Use the itegral test to determie whether the series = l() coverges or diverges. Show your work ad clearly state your aswer. Solutio: Let f (x) = x l x. f (x) = l x + x (l x) ) decreasig. Let u = l x the du = dx x ad it dx = lim x l x M M M = lim M l = lim M l u M l = diverges < 0 for x > so the fuctio is x l x dx u du Sice the itegral diverges, the the series also diverges. 7. (4 poits) Use the compariso test to determie whether the series coverges or diverges. l k k k= Solutio: Let b k = /k ad a k = (l k)/k. For k, l k so a k b k. Sice k= b k is the (diverget) haroic series (that is, the p-series with p = ), this series diverges. Page of
MA 4 Exam Fall 06 8. A fuctio f is defied by f (x) = + x + x + 4 4 x + + + + + x + = =0 + x. for all x i the iterval of covergece for the power series. (a) (4 poits) Fid the radius of covergece for the power series. Show your work. Solutio: Apply the ratio test with We get a = + + x. lim a + a = lim + + + + x = x so the radius of covergece is R =. (b) (4 poits) Fid the iterval of covergece for the power series. work. Show your Solutio: We check x = ±. If x = we get the umerical series + =0 + ( ) = =0 ( ) ( + ). This series diverges by the diverget series test sice lim ( ) ( + ) does ot exist. For x = + a similar aalysis shows that the series diverges. Hece, the iterval of covergece is (, ). Page of
MA 4 Exam Fall 06 (c) (4 poits) Fid the power series represetatio for f (x) ad state its radius of covergece. Solutio: We ca differetiate term-by-term to get + + x = ( + ) =0 + + x which coverges absolutely for x <. (d) (4 poits) Fid the power series represetatio for f (x) dx. Solutio: We ca itegrate term by term ad get which coverges for x <. + x + ( x ) + =0 + + + C = + C =0 Page of