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)

(B)

(C)

(D)

(E)

2.

(A) s_n diverges by oscillation

(B) s_n converges to zero

(C)

(D) s_n diverges to infinity

(E) None of the above is true.

3. The sequence

(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)

(B)

(C)

(D)

(E)

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

(A) |r| < 1

(B) |r| 1

(C) −1 < r 1

(D) 0 < r < 1

(E) |r| > 1

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

(A) converges to a finite sum.

(B) equals zero.

(C) does not diverge to infinity.

(D) is a positive series.

(E) none of these

7. Note that equals

(A) 0

(B) 1

(C)

(D)

(E) ∞

8. The sum of the geometric series

(A)

(B)

(C) 1

(D)

(E)

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

(A) If converges.

(B) If diverges.

(C) If diverges, then

(D) converges if and only if

(E) none of these

10. Which of the following series diverges?

(A)

(B)

(C)

(D)

(E) none of these

11. Which of the following series diverges?

(A)

(B)

(C)

(D) 1−1.1 + 1.21−1.331 + ···

(E)

12. Let then S equals

(A) 1

(B)

(C)

(D) 2

(E) 3

13. Which of the following expansions is impossible?

(A) in powers of x

(B) in powers of x

(C) ln x in powers of (x − 1)

(D) tan x in powers of

(E) ln (1 − x) in powers of x

14. The series converges if and only if

(A) x = 0

(B) 2 < x < 4

(C) x = 3

(D) 2 x 4

(E) x < 2 or x > 4

15. Let The radius of convergence of is

(A) 0

(B) 1

(C) 2

(D) ∞

(E) none of these

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

(A)

(B)

(C)

(D)

(E)

17. If an appropriate series is used to evaluate 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 is used to approximate 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) where m is any positive integer.

(B) If converges, so does if c ≠ 0.

(C) If and converge, so does 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)

(B)

(C)

(D)

(E)

23. Which of the following series diverges?

(A)

(B)

(C)

(D)

(E)

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

(A)

(B)

(C)

(D)

(E)

25. Which of the following alternating series diverges?

(A)

(B)

(C)

(D)

(E)

26. Which of the following statements is true?

(A) If converges, then so does the series

(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 converges.

(E) none of these

27. The power series converges if and only if

(A) −1 < x < 1

(B) −1 x 1

(C) −1 x < 1

(D) −1 < x 1

(E) x = 0

28. The power series

diverges

(A) for no real x

(B) if −2<x 0

(C) if x < −2 or x > 0

(D) if −2 x < 0

(E) if x ≠ −1

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

converges for

(A) 1 x 3

(B) 1 x < 3

(C) 1 < x 3

(D) 0 x 4

(E) none of these

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

(A)

(B)

(C)

(D)

(E)

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

(A)

(B)

(C)

(D)

(E)

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

(A)

(B)

(C)

(D)

(E)

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 e^{sin x} is

(A) 0

(B) 1

(C)

(D) −1

(E)

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

(A) converges to f (x₀) + g(x₀).

(B) converges to f (x₀)g(x₀).

(C) is continuous at x = x₀.

(D) 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)

(B)

(C)

(D)

(E)

38. The radius of convergence of the series

(A) 0

(B) 2

(C)

(D)

(E) ∞

39. If the approximate formula sin x = 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 correct to three decimal places, is

(A) −0.200

(B) 0.180

(C) 0.190

(D) −0.190

(E) −0.990

41. The function 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 is equal to

(A) 0

(B) 1

(C)

(D)

(E) none of these

43. When 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)

(C)

(D)

(E) none of these

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

(A)

(B) 2x − 2x² + 8x³ − 16x⁴ + · · ·

(C) 2x − 4x² + 16x³ + · · ·

(D)

(E)

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

(A)

(B)

(C)

(D)

(E)

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)