MATH 152, SPRING 2006 COMMON EXAM III - VERSION B LAST NAME, First Name (print): INSTRUCTOR: SECTION NUMBER: UIN: DIRECTIONS: 1. The use of a calculator, laptop or computer is prohibited. 2. In Part 1 (Problems 1-10), mark the correct choice on your ScanTron form No. 815-E using a No. 2 pencil. For your own records, also record your choices on your exam! ScanTrons will be collected from all examinees after 90 minutes and will not be returned. 3. In Part 2 (Problems 11-15), present your solutions in the space provided. Show all your work neatly and concisely and clearly indicate your nal answer. You will be graded not merely on the nal answer, but also on the quality and correctness of the work leading up to it. 4. Be sure to write your name, section number and version letter of the exam on the ScanTron form. THE AGGIE CODE OF HONOR \An Aggie does not lie, cheat or steal, or tolerate those who do." Signature: DO NOT WRITE BELOW! Question Points Awarded Points 1-10 50 11 10 12 15 13 6 14 9 15 10 100 1 PART I 1. (5 pts) lim n!1 n 2 +1 n 2 sin n 2n +1 = (a) −2 (b) 1 (c) 2 (d) doesn’t exist (e) −1 2. (5 pts) Which series converges, but not absolutely? I. 1 X n=1 (−1) n n 8 II. 1 X n=1 (−1) n+1 n 5 +2 n 3 III. 1 X n=1 (−1) n 4 p n (a) only III (b) I and II (c) II and III (d) I and III (e) only I Exam continues on next page 2 3. (5 pts) The Maclaurin series for 1=(1 + x 4 )is (a) 1 X n=0 x −4n (b) 1 X n=0 (−1) n x 2n (c) 1 X n=0 (−1) n x 4n (d) 1 X n=0 x 4n (e) 1 X n=0 x 2n 4. (5 pts) Compute 1 X n=1 1 n(n +1) . (Hint: Use partial fractions) (a) 1 (b) 7 6 (c) 2 (d) 5 6 (e) 4 5 Exam continues on next page 3 5. (5 pts) Which of the following are true? I. If lim n!1 b n =0,then 1 X n=1 b n converges. II. If 0 a n b n and 1 X n=1 b n diverges, then 1 X n=1 a n diverges. III. If 1 X n=1 a n converges, then lim n!1 a n =0. (a) only I (b) I and III (c) only II (d) II and III (e) only III 6. (5 pts) 1 X n=1 2 n 3 n = (a) Does not exist (b) 2 (c) 3 (d) 2 3 (e) 3 2 Exam continues on next page 4 7. (5 pts) 1 X n=3 1 n ln n (a) diverges by the integral test (b) converges by the integral test (c) converges by the ratio test (d) diverges by the ratio test (e) converges by the alternating series test 8. (5 pts) The sequence sinn n 1 n=1 (a) converges to 0 (b) converges, but is not bounded (c) converges to 1 (d) diverges, but is bounded (e) none of the above Exam continues on next page 5 9. (5 pts) For what values of x does the power series 1 X n=1 x 2n p n converge? (a) for all x (b) only x =0 (c) −1 x<1 (d) −1 x 1 (e) −1 1 (c) diverges for all p (d) converges for all p (e) converges if p>0 Exam continues on next page 6 PART II 11. (10 pts) Find the radius and interval of convergence for the power series 1 X n=1 (x − 3) n n4 n . Identify the test(s) you are using and clearly show your work. Exam continues on next page 7 12. Determine whether the in nite series converge or diverge. Identify the test(s) you are using and clearly show your work. (a) (5 pts) 1 X n=1 e n n! (b) (5 pts) 1 X n=1 (−1) n+1 1+lnn (c) (5 pts) 1 X n=1 n n 2:001 +4 Exam continues on next page 8 13. (6 pts) Find the Taylor series for f(x)=e 3x at a =2. 14. (a) (5 pts) Use the Maclaurin series for sin z to express R 1 0 sin(x 2 )dx as an in nite series. (b) (4 pts) Find a bound on the error made when approximating 1 X n=1 (−1) n+1 n 2 by 9 X n=1 (−1) n+1 n 2 . Exam continues on next page 9 15. (a) (5 pts) Find two distinct unit vectors that are parallel to h1;2;−2i. (b) (5 pts) Find the vector projection of b = h4; 2; 0i onto a = h1;−1;1i. End of exam 10