AP Calculus Convergence of Series Student Handout 2016-2017 EDITION Use the following link or scan the QR code to complete the evaluation for the Study Session https://www.surveymonkey.com/r/S_SSS Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org Convergence of Series Students should be able to: Recognize various types of numerical series and efficiently apply the appropriate test. Understand that a series may be absolutely convergent, conditionally convergent or divergent and utilize proper techniques to decide. Determine the sum of an infinite geometric series and be able to use that sum to create a power series and determine its interval of convergence. Understand that an infinite series of numbers converges to a real number S (or has sum S), if and only if the limit of its sequence of partial sums exists and equals to S. Use the methods the nth term test, the comparison test, the limit comparison test, the geometric series test, p-series test, the integral test, the ratio test and the alternating series test for determining whether the series of numbers converges or diverges. Use the ratio test to determine radius or open interval of convergence of power series. Use the other tests to check convergence at the endpoints. Use the alternating series error bound, if an alternating series converges, to estimate how close a partial sum is to the value of the infinite series. Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 1 Multiple Choice 1. (calculator not allowed) n n 2 n What is the interval of convergence for the power series 1 n x 4 ? n0 n 3 (A) 3 x 3 (B) 3 x 3 (C) 1 x 7 (D) 1 x 7 2. (calculator not allowed) Which of the following series are convergent? 1 1 1 I. 1 . . 2 2 32 n 2 1 1 1 II. 1 . . 2 3 n 1 1 1 n1 III. 1 . . 3 32 3n1 (A) I only (B) III only (C) I and III only (D) II and III only (E) I, II, and I 3. (calculator not allowed) The Taylor series for a function f about x 0 converges to f for 1 x 1. The nth-degree n k k x Taylor Polynomial for f about x 0 is given by Pn x 1 2 . Of the following, k1 k k 1 which is the smallest number M for which the alternating series error bound guarantees that f 1 P4 1 M ? 1 1 (A) 5! 31 1 1 (B) 4! 21 1 (C) 31 1 (D) 21 Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 2 4. (calculator not allowed) 1n Which of the following statements about the series is true? n1 1 n (A) The series converges absolutely. (B) The series converges conditionally. (C) The series converges but neither conditionally nor absolutely. (D) The series diverges. 5. (calculator not allowed) Which of the following series can be used with the limit comparison test to determine whether the n series 3 converges or diverges? n1 n 1 1 (A) n1 n 1 (B) 2 n1 n 1 (C) 3 n1 n n3 1 (D) 2 n1 n 6. (calculator not allowed) Which of the following series are conditionally convergent? n n 1 1 n 1 I. II. III. 4 3 2 n1 n n1 n n1 n (A) III only (B) I and II (C) II and III (D) I, II and III Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 3 7. (calculator not allowed) Which of the following series is absolutely convergent? n1 1 (A) 1 n 1 2n n1 1 (B) 1 n 1 n n1 n (C) 1 n 1 n 1 n n1 1 (D) 1 n 1 2 8. (calculator not allowed) Which of the following series converge? n I. n 1 n 2 cosn II. n 1 n 1 III. n 1 n (A) None (B) II only (C) III only (D) I and II only (E) I and III only Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 4 9. (calculator not allowed) b dx If lim is finite, then which of the following must be true? b 1 x p 1 (A) converges p n 1 n 1 (B) diverges p n 1 n 1 (C) converges p2 n 1 n 1 (D) converges p 1 n 1 n 1 (E) diverges p 1 n 1 n 10. (calculator not allowed) 2n 1 What is the value of ? n n 1 3 (A) 1 (B) 2 (C) 4 (D) 6 (E) The series diverges 11. (calculator not allowed) n What are all values of p for which the infinite series converges? p n 1 n 1 (A) p 0 (B) p1 (C) p1 (D) p 2 (E) p 2 Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 5 12. (calculator not allowed) Which of the following series diverge? n sin 2 I. n 0 1 II. 3 n 1 n en III. n n 1 e 1 (A) III only (B) I and II only (C) I and III only (D) II and III only (E) I, II, and III 13. (calculator not allowed) en Consider the series . If the ratio test is applied to the series, which of the following inequalities n1 n! results, implying that the series converges? e (A) lim 1 n n! n! (B) lim 1 n e n 1 (C) lim 1 n e e (D) lim 1 n n 1 e (E) lim 1 n (1)!n Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 6 14. (calculator not allowed) Which of the following series converges for all real numbers x ? xn (A) n1 n xn (B) 2 n1 n xn (C) n1 n exnn (D) n1 n! nx! n (E) n n1 e 15. (calculator not allowed) 2 n What are all values of x for which the series converges? 2 n1 x 1 (A) 11x (B) x 1 only (C) x 1 only (D) x 1 and x 1 only (E) x 1 and x 1 16. (calculator not allowed) n 1 p For what values of p will both series and converge? 2 p n 1 n n 1 2 (A) 2 p 2 1 1 (B) p 2 2 1 (C) p 2 2 1 (D) p and p 2 2 (E) There are no such values of p. Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 7 17. (calculator not allowed) For x 1, the continuous function g is decreasing and positive. A portion of the graph of g is shown above. For n 1, the nth term of the series a is defined by a g(n). If g(x)dx n n n 1 1 converges to 8, which of the following could be true? (A) an 6 n 1 (B) an 8 n 1 (C) an 10 n 1 (D) diverges an n 1 18. (calculator allowed) The power series n converges conditionally at . Which of the following bn x 2 x 1 n 1 statements about the convergence of the series at x 6is true? (A) The series converges conditionally at x 6 (B) The series diverges at x 6 (C) The series converges absolutely at x 6 (D) The convergence at x 6 cannot be determined using the given information Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 8 19. (calculator not allowed) n x The Maclaurin series for the function f is given by f (x) . What is the value of n0 4 f (3)? (A) 3 3 (B) 7 4 (C) 7 13 (D) 16 (E) 4 20. (calculator allowed) If the series an converges and an 0 for all n, which of the following must be true? n1 a (A) lim n1 0 n an (B) an 1 for all n (C) an 0 n1 (D) nan diverges. n1 a (E) n converges. n1 n 21. (calculator not allowed) n The infinite series ak has nth partial sum Sn for n 1. What is the sum of the k1 1 4n series ak ? k1 (A) The series diverges. 1 (B) 4 1 (C) 4 1 (D) 3 Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 9 22. (calculator not allowed) 2 4 6 8 Find the sum of the series 1 2! 4! 6! 8! (A) -1 (B) 0 (C) 1 (D) The series diverges Free Response 23. (calculator not allowed) 1n 1 (c) Give a value of p such that converges, but diverges. Give reasons p 2 p n1 n n 1 n why your value of p is correct. 1 1 (d) Give a value of p such that diverges, but converges. Give reasons why p 2 p n1 n n 1 n your value of p is correct. Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 10 24.