Practice Exercises

*A1. Which sequence converges?

*A2. If , then

(A) s_n diverges by oscillation
(B) s_n converges to zero
(C) 
(D) s_n diverges to infinity

*A3. The sequence 

(A) is unbounded
(B) converges to a number less than 1
(C) is bounded
(D) diverges to infinity

*A4. Which of the following sequences diverges?

*A5. The sequence {rⁿ} converges if and only if

(A) |r| < 1
(B) |r| ≤ 1
(C) –1 < r ≤ 1
(D) 0 < r < 1

*A6.  is a series of constants for which . Which of the following statements is always true?

(A)  converges to a finite sum.
(B)  does not diverge to infinity.
(C)  is a positive series.
(D) None of statements (A), (B), or (C) are always true.

*A7. Note that .  equals

*A8. The sum of the geometric series  is

*A9. Which of the following statements about series is true?
(A) If , then  converges.
(B) If , then  diverges.
(C) If  diverges, then .
(D)  converges if and only if .

*A10. Which of the following series diverges?

*A11. Which of the following series diverges?

*A12. Let ; then S equals

*A13. 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 

*A14. The series  converges if and only if

(A) x = 0
(B) 2 < x < 4
(C) x = 3
(D) 2 ≤ x ≤ 4

*A15. Let . The radius of convergence of  is

(A) 0
(B) 1
(C) 2
(D) e

*A16. The coefficient of x⁴ in the Maclaurin series for f(x) = e^–x/2 is

*A17. If an appropriate series is used to evaluate , then, correct to four decimal places, the definite integral equals

(A) 0.0002
(B) 0.0003
(C) 0.0032
(D) 0.0033

*A18. 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) 200
(B) 300
(C) 400
(D) 500

*A19. Let f(x) = tan^–1x and P₇(x) be the Taylor polynomial of degree 7 for f about x = 0. Given , it follows that if –0.5 < x < 0.5,

(A) P₇(x) ≤ tan^–1x
(B) P₇(x) ≥ tan^–1x
(C) P₇(x) > tan^–1x if x < 0 but P₇(x) < tan^–1x if x > 0
(D) P₇(x) < tan^–1x if x < 0 but P₇(x) > tan^–1x if x > 0

*A20. Let f(x) = tan^–1x and P₉(x) be the Taylor polynomial of degree 9 for f about x = 0. Given , it follows that if –0.5 < x < 0.5,

(A) P₉(x) ≤ tan^–1x
(B) P₉(x) ≥ tan^–1 x
(C) P₉(x) > tan^–1x if x < 0 but P₉(x) < tan^–1x if x > 0
(D) P₉(x) < tan^–1x if x < 0 but P₉(x) > tan^–1x if x > 0

*A21. Which of the following series converges?

*A22. Which of the following series diverges?

*A23. For which of the following series does the Ratio Test fail?

*A24. Which of the following alternating series diverges?

*A25.The power series  converges if and only if

(A) –1 < x < 1
(B) –1 ≤ x ≤ 1
(C) –1 ≤ x < 1
(D) –1 < x ≤ 1

*A26. The power series  diverge

(A) for no real x
(B) if –2 < x ≤ 0
(C) if x < –2 or x > 0
(D) if –2 ≤ x < 0

*A27. 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

*A28. The Taylor polynomial of degree 3 at x = 0 for  is

*A29. The Taylor polynomial of degree 3 at x = 1 for e^x is

*A30. The coefficient of  in the Taylor series about  of f(x) = cos x is

*A31. The coefficient of x² in the Maclaurin series for e^{sin x} is

*A32. The coefficient of (x – 1)⁵ in the Taylor series for x ln x about x = 1 is

A33. The radius of convergence of the series  is

*A34. The Taylor polynomial of degree 3 at x = 0 for (1 + x)^p, where p is a constant, is

*A35. The Taylor series for ln (1 + 2x) about x = 0 is

*A36. The set of all values of x for which  converges is

(A) only x = 0
(B) –2 < x < 2
(C) |x| > 2
(D) |x| ≥ 2

*A37. The third-degree Taylor polynomial P₃(x) for sin x about  is

*A38. 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

*B1. 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) Rearranging the terms of a positive convergent series will not affect its convergence or its sum.

*B2. Which of the following statements is always 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) None of statements (A), (B), or (C) are always true.

*B3. 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

*B4. Let . Suppose both series converge for |x| < R. Let x₀ be a number such that |x₀| < R. Which of statements (A)–(C) is false?

(A)  converges to f(x₀) + g(x₀).
(B)  is continuous at x = x₀.
(C)  converges to f′(x₀).
(D) Statements (A)–(C) are all true.

*B5. If the approximate formula  is used and |x| < 1 (radian), then the error is numerically less than

(A) 0.003
(B) 0.005
(C) 0.008
(D) 0.009

*B6. The function  and f′(x) = –f(x) for all x. If f(0) = 1, then how many terms of the series are needed to find f(0.2) correct to three decimal places?

(A) 2
(B) 3
(C) 4
(D) 5 

*B7. The sum of the series 

*B8. 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

*B9. You wish to estimate e^x, over the interval |x| ≤ 2, with an error less than 0.001. The Lagrange error term suggests that you use a Taylor polynomial at 0 with degree at least

(A) 9
(B) 10
(C) 11
(D) 12