SCORE. Exam 2. MA 114 Exam 2 Fall 2016

Transcription

1 Exam 2 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 2 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 2 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 2 A B C D E SCORE Multiple Total Choice Score

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3 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 ( + )! < 2. The iterval of covergece of the power series A. [0] ( B., ) C. (, ] D. (, ) E. (, +) ( x ) is =0 Page of 2

4 . The sum of the ifiite geometric series is A. 5 B. 2 C. 5 D. 2 E Which of the followig sequeces coverge? { } 5 I. 2 { 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 2

5 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 2 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 2

6 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 Let S N be the N-th partial sum of the series Thus, S =, S 2 = 2. Compute S 50 S 49. A. 99. B. 50 C. 2 D. 960 E. 0 ( ) 2. Page 6 of 2

7 9. Cosider the series leads to the followig coclusio. 4 A. The test is icoclusive. B. The series coverges absolutely. 4. Applyig the compariso test with the series C. The series coverges coditioally. D. The series diverges. E. The test caot be applied to a = ad b = The radius of covergece for the series A. B. /0 C. 0 D. /0 E. 2 x =0 0 is Page 7 of 2

8 . The series 2 + =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.. 2. cos(π) 2. The series 2 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 2

9 Free Respose Questios. Fid the first four (4) terms of each of the followig sequeces. (a) (6 poits) a = ( + )! (b) (6 poits) a = 2 ad a + = a 4. Determie if the sequece is coverget or diverget. If coverget give its limit. (a) (4 poits) a = + (b) (4 poits) a = 2 e (c) (4 poits) a = 2 Page 9 of 2

10 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) (b) (5 poits) si 2 () (c) (5 poits) 2 2 Page 0 of 2

11 6. (5 poits) Use the itegral test to determie whether the series =2 l() coverges or diverges. Show your work ad clearly state your aswer. 7. (4 poits) Use the compariso test to determie whether the series coverges or diverges. l k k k= Page of 2

12 8. A fuctio f is defied by f (x) = x + x 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. (b) (4 poits) Fid the iterval of covergece for the power series. work. Show your (c) (4 poits) Fid the power series represetatio for f (x) ad state its radius of covergece. (d) (4 poits) Fid the power series represetatio for f (x) dx. Page 2 of 2