MATH 4 EXAMINATION II MARCH 24, 2004 TEST FORM A NAME STUDENT NUMBER INSTRUCTOR SECTION NUMBER This examination consists of 2 problems. The first 6 are multiple choice questions, the next two are short answer questions and the remaining 4 are partial credit problems. The point value for each question appears to the left of the question number. There are 00 total points. Please record your answers to the multiple choice questions by circling the corresponding letter. Present your work clearly for the partial credit problems. No credit will be given for unsupported answers. THE USE OF CALCULATORS, BOOKS, NOTES ETC. DURING THIS EXAMINATION IS PROHIBITED. Do not write in the blanks below.. (5) 7. (2) 2. (5) 8. (0) 3. (5) 9. (2) 4. (5) 0. (8) 5. (5). (8) 6. (5) 2. (0) SUBTOTAL TOTAL
MATH 4 EXAMINATION II, TEST FORM A PAGE 2. Find the sum of the series n=2 ( ) n 2 4 n a) 24 b) 0 c) d) 5 40 e) The series diverges ( sin 2 n ) n, 2. If a n = then the sequence {an } 2 a) diverges by oscillation b) to 0 c) to d) to π 2 e) diverges to +
MATH 4 EXAMINATION II, TEST FORM A PAGE 3 3. The series to n= ( ) tan n tan (n + ) a) π 4 b) c) 0 d) π 2 e) It does not converge, it diverges to. 4. When evaluated, the integral a) to 0. b) to. c) to e. d) diverges to. e) diverges to +. + 0 xe x dx
MATH 4 EXAMINATION II, TEST FORM A PAGE 4 5. Given the series ( ) n+, n! n= determine the smallest n that will allow s n, the n th partial sum, to approximate the actual sum to within 00. a) 2 b) 4 c) 6 d) 8 e) 0 6. The series n= n 3 n 4 + a) Converges by the Direct Comparison Test with b) by the nth term Test for Divergence. n= n. c) by the nth Root Test. d) Diverges by the Limit Comparison Test with e) diverges by the Ratio Test. n= n.
MATH 4 EXAMINATION II, TEST FORM A PAGE 5 2 pts 7. If an is a convergent series with all positive terms, i.e.., a n > 0 for all n, then determine which of the following must always be true. Circle the correct answer that corresponds to each question. NOTE: You do not need to justify your work. a) cos an Must Must Can not be converge diverge determined b) a n Must Must Can not be converge diverge determined c) an Must Must Can not be converge diverge determined d) (an ) 2 Must Must Can not be converge diverge determined
MATH 4 EXAMINATION II, TEST FORM A PAGE 6 0 pts 8. Determine whether the following sequences converge or diverge. For credit, circle your answer choice and find the limit of each convergent sequence. NOTE: You do not need to show your work. { } 3n2 + 2 2 pts a) 2n + 3 Diverges or Converges to 2 pts b) { 2 3, 4 5, 6 7, 8 } 9,..., ( )n 2n 2n +,... Diverges or Converges to 2 pts c) { } n! n n Diverges or Converges to 2 pts d) { cos 5 } n Diverges or Converges to 2 pts e) { n ( n) } Diverges or Converges to
MATH 4 EXAMINATION II, TEST FORM A PAGE 7 2 pts 9. Determine whether the following series converge absolutely, converge conditionally, or diverge. Circle the correct answer that corresponds to each question. NOTE: You do not need to justify your work. a) n= cos n n 2 Absolutely Conditionally Diverges b) n=0 n! ( 2) n Absolutely Conditionally Diverges c) n=2 ( ) n+ n Absolutely Conditionally Diverges d) n=2 cos(πn) 2n Absolutely Conditionally Diverges
MATH 4 EXAMINATION II, TEST FORM A PAGE 8 8 pts 0. Evaluate 4 0 dx. For credit, you must show all work that leads to your answer. (x 2) 3
MATH 4 EXAMINATION II, TEST FORM A PAGE 9 8 pts. Test the series n=4 n 3 n 3n 4 + 5 n + for convergence or divergence. At the bottom of the page, name the tests or theorems used and fully justify your conclusion. No credit will be given for unsupported answers. Test(s) or theorem(s) used Conclusion
MATH 4 EXAMINATION II, TEST FORM A PAGE 0 0 pts 2. Determine whether the series n=2 ( ) n ln n n absolutely, conditionally, or diverges. At the bottom of the page, name the tests or theorems used and fully justify your conclusion. No credit will be given for unsupported answers. You must give a full justification of your answer. Test(s) or theorem(s) used Conclusion