Calculus AB and Calculus BC

CHAPTER 10 Sequences and Series

Practice Exercises

Part A. Directions: Answer these questions without using your calculator.

Note: No questions on sequences will appear on the BC examination. We have nevertheless chosen to include the topic in Questions 1–5 because a series and its convergence are defined in terms of sequences. Review of sequences will enhance understanding of series.

1. Which sequence converges?

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

2. Image

(A) sn diverges by oscillation

(B) sn converges to zero

(C) Image

(D) sn diverges to infinity

(E) None of the above is true.

3. The sequence Image

(A) is unbounded

(B) is monotonic

(C) converges to a number less than 1

(D) is bounded

(E) diverges to infinity

4. Which of the following sequences diverges?

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

5. The sequence {rn } converges if and only if

(A) |r| < 1

(B) |r| Image 1

(C) −1 < r Image 1

(D) 0 < r < 1

(E) |r| > 1

6. Image is a series of constants for which Image Which of the following statements is always true?

(A) Image converges to a finite sum.

(B) Image equals zero.

(C) Image does not diverge to infinity.

(D) Image is a positive series.

(E) none of these

7. Note that Image equals

(A) 0

(B) 1

(C) Image

(D) Image

(E)

8. The sum of the geometric series Image

(A) Image

(B) Image

(C) 1

(D) Image

(E) Image

9. Which of the following statements about series is true?

(A) If Image converges.

(B) If Image diverges.

(C) If Image diverges, then Image

(D) Image converges if and only if Image

(E) none of these

10. Which of the following series diverges?

(A) Image

(B) Image

(C) Image

(D) Image

(E) none of these

11. Which of the following series diverges?

(A) Image

(B) Image

(C) Image

(D) 1−1.1 + 1.21−1.331 + ···

(E) Image

12. Let Image then S equals

(A) 1

(B) Image

(C) Image

(D) 2

(E) 3

13. Which of the following expansions is impossible?

(A) Image in powers of x

(B) Image in powers of x

(C) ln x in powers of (x − 1)

(D) tan x in powers of Image

(E) ln (1 − x) in powers of x

14. The series Image converges if and only if

(A) x = 0

(B) 2 < x < 4

(C) x = 3

(D) 2 Image x Image 4

(E) x < 2 or x > 4

15. Let Image The radius of convergence of Image is

(A) 0

(B) 1

(C) 2

(D)

(E) none of these

16. The coefficient of x4 in the Maclaurin series for f (x) = ex/2 is

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

17. If an appropriate series is used to evaluate Image then, correct to three decimal places, the definite integral equals

(A) 0.009

(B) 0.082

(C) 0.098

(D) 0.008

(E) 0.090

18. If the series tan−1 Image is used to approximate Image with an error less than 0.001, then the smallest number of terms needed is

(A) 100

(B) 200

(C) 300

(D) 400

(E) 500

19. Let f be the Taylor polynomial P7 (x) of order 7 for tan−1 x about x = 0. Then it follows that, if −0.5 < x < 0.5,

(A) f (x) = tan−1 x

(B) f (x) tan−1 x

(C) f (x) tan−1 x

(D) f (x) > tan−1 x if x < 0 but < tan−1 x if x > 0

(E) f (x) < tan−1 x if x < 0 but > tan−1 x if x > 0

20. Replace the first sentence in Question 19 by “Let f be the Taylor polynomial P9 (x) of order 9 for tan−1 x about x = 0.” Which choice given in Question 19 is now the correct one?

Part B. Directions: Some of the following questions require the use of a graphing calculator.

21. Which of the following statements about series is false?

(A) Image where m is any positive integer.

(B) If Image converges, so does Image if c ≠ 0.

(C) If Image and Image converge, so does Image where c ≠ 0.

(D) If 1000 terms are added to a convergent series, the new series also converges.

(E) Rearranging the terms of a positive convergent series will not affect its convergence or its sum.

22. Which of the following series converges?

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

23. Which of the following series diverges?

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

24. For which of the following series does the Ratio Test fail?

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

25. Which of the following alternating series diverges?

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

26. Which of the following statements is true?

(A) If Image converges, then so does the series Image

(B) If a series is truncated after the nth term, then the error is less than the first term omitted.

(C) If the terms of an alternating series decrease, then the series converges.

(D) If r < 1, then the series Image converges.

(E) none of these

27. The power series Image converges if and only if

(A) −1 < x < 1

(B) −1 Image x Image 1

(C) −1 Image x < 1

(D) −1 < x Image 1

(E) x = 0

28. The power series

Image

diverges

(A) for no real x

(B) if −2<x Image 0

(C) if x < −2 or x > 0

(D) if −2 Image x < 0

(E) if x ≠ −1

29. The series obtained by differentiating term by term the series

Image

converges for

(A) 1 Image x Image 3

(B) 1 Image x < 3

(C) 1 < x Image 3

(D) 0 Image x Image 4

(E) none of these

30. The Taylor polynomial of order 3 at x = 0 for Image is

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

31. The Taylor polynomial of order 3 at x = 1 for ex is

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

32. The coefficient of Image in the Taylor series about Image of f (x) = cos x is

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

33. Which of the following series can be used to compute ln 0.8?

(A) ln (x − 1) expanded about x = 0

(B) ln x about x = 0

(C) ln x expanded about x = 1

(D) ln (x − 1) expanded about x = 1

(E) none of these

34. If e−0.1 is computed using a Maclaurin series, then, correct to three decimal places, it equals

(A) 0.905

(B) 0.950

(C) 0.904

(D) 0.900

(E) 0.949

35. The coefficient of x2 in the Maclaurin series for esin x is

(A) 0

(B) 1

(C) Image

(D) −1

(E) Image

36. Let Image Suppose both series converge for |x| < R. Let x0 be a number such that |x0 | < R. Which of the following statements is false?

(A) Image converges to f (x0) + g(x0).

(B) Image converges to f (x0)g(x0).

(C) Image is continuous at x = x0.

(D) Image converges to f (x0).

(E) none of these

37. The coefficient of (x − 1)5 in the Taylor series for x ln x about x = 1 is

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

38. The radius of convergence of the series Image

(A) 0

(B) 2

(C) Image

(D) Image

(E)

39. If the approximate formula sin x = Image is used and |x| < 1 (radian), then the error is numerically less than

(A) 0.001

(B) 0.003

(C) 0.005

(D) 0.008

(E) 0.009

40. If a suitable series is used, then Image correct to three decimal places, is

(A) −0.200

(B) 0.180

(C) 0.190

(D) −0.190

(E) −0.990

41. The function Image and f (x) = −f (x) for all x. If f (0) = 1, then f (0.2), correct to three decimal places, is

(A) 0.905

(B) 1.221

(C) 0.819

(D) 0.820

(E) 1.220

42. The sum of the series Image is equal to

(A) 0

(B) 1

(C) Image

(D) Image

(E) none of these

43. When Image is approximated by the sum of its first 300 terms, the error is closest to

(A) 0.001

(B) 0.002

(C) 0.005

(D) 0.01

(E) 0.02

44. The Taylor polynomial of order 3 at x = 0 for (1 + x)p, where p is a constant, is

(A) 1 + px + p(p − 1)x2 + p(p − 1)(p − 2)x3

(B) Image

(C) Image

(D) Image

(E) none of these

45. The Taylor series for ln (1 + 2x) about x = 0 is

(A) Image

(B) 2x − 2x2 + 8x3 − 16x4 + · · ·

(C) 2x − 4x2 + 16x3 + · · ·

(D) Image

(E) Image

46. The set of all values of x for which Image converges is

(A) only x = 0

(B) |x| = 2

(C) −2 < x < 2

(D) |x| > 2

(E) none of these

47. The third-order Taylor polynomial P3 (x) for sin x about Image is

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

48. Let h be a function for which all derivatives exist at x = 1. If h(1) = h′ (1) = h″ (1) = h′″ (1) = 6, which third-degree polynomial best approximates h there?

(A) 6 + 6x + 6x2 + 6x3

(B) 6 + 6(x − 1) + 6(x − 1)2 + 6(x − 1)3

(C) 6 + 6x + 3x2 + x3

(D) 6 + 6(x − 1) + 3(x − 1)2 + (x − 1)3

(E) Image