152, MATH 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 final answer. You will be graded not merely on the final 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.
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DO NOT WRITE BELOW! Question 1-10 11 12 13 14 15 Points Awarded Points 50 10 15 6 9 10 100
1
PART I
1. (5 pts) lim (a) -2 (b) 1 (c) 2
n
n2 + 1 sin n2
n 2n + 1
=
(d) doesn't exist (e) -1
2. (5 pts) Which series converges, but not absolutely? (-1)n n8 n=1
I.
II.
n=1
(-1)n+1
n5 + 2 n3
III.
(-1)n 4 n n=1
(a) only III (b) I and II (c) II and III (d) I and III (e) only I
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3. (5 pts) The Maclaurin series for 1/(1 + x4 ) is
(a)
n=0
x-4n (-1)n x2n (-1)n x4n x4n x2n
(b)
n=0
(c)
n=0
(d)
n=0
(e)
n=0
4. (5 pts) Compute (a) 1 7 (b) 6 (c) 2 5 (d) 6 4 (e) 5
1 . (Hint: Use partial fractions) n(n + 1) n=1
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5. (5 pts) Which of the following are true?
I. If lim bn = 0 , then
n n=1
bn converges.
II. If 0 an bn and
n=1
bn diverges, then
n=1
an diverges.
III. If
n=1
an converges, then lim an = 0 .
n
(a) I only (b) I and III (c) only II (d) II and III (e) only III
6. (5 pts)
2n = 3n n=1
(a) Does not exist (b) 2 (c) 3 2 (d) 3 3 (e) 2
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7. (5 pts)
1 n ln n n=3
(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 (a) converges to 0
sin n n
n=1
(b) converges, but is not bounded (c) converges to 1 (d) diverges, but is bounded (e) none of the above
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9. (5 pts) For what values of x does the power series (a) for all x (b) only x = 0 (c) -1 x < 1 (d) -1 x 1 (e) -1 < x < 1
x2n converge? n n=1
10. (5 pts)
1 (1 + n)np n=1
(a) diverges for all p 1 (b) diverges if p > 1 (c) diverges for all p (d) converges for all p (e) converges if p > 0
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PART II (x - 3)n . Identify the test(s) you are n 4n n=1
11. (10 pts) Find the radius and interval of convergence for the power series using and clearly show your work.
Exam continues on n...