MATH 137 : Calculus 1 for Honours Mathematics Instructor: Barbara Forrest Self Check #1: Sequences and Convergence

Size: px

Start display at page:

Download "MATH 137 : Calculus 1 for Honours Mathematics Instructor: Barbara Forrest Self Check #1: Sequences and Convergence"

Luke Gallagher
5 years ago
Views:

1 1 MATH 137 : Calculus 1 for Honours Mathematics Instructor: Barbara Forrest Self Check #1: Sequences and Convergence Recommended Due Date: complete by the end of WEEK 3 Weight: none - self graded INSTRUCTIONS: Ensure you have completed the daily/weekly tasks up to and including Monday, Week 3 on the Introduction to Series to the end of Chapter 1 of the Course Notes. Read and complete the following self check assignment on your own. Write your answers on the printed copy of this assignment. DO NOT SUBMIT THIS ASSIGNMENT Students are encouraged to complete this assignment on their own. Do not submit this assignment to your instructor. As you complete this assignment, you should consider it similar to an open-book chapter exam. OBJECTIVES The purpose of this assignment is for the student to check if they have understood the concepts covered in the course. Though this assignment is not for credit, students are encouraged to complete it since questions on the DROPBOX ASSIGNMENTS and the FINAL EXAM will be the same or similar to those found on this self check. Full solutions will be posted on the course website so that you can check your work, view how your instructor expects you to present solutions, and to see if you understand the topics in the first chapter of the course notes. You can use Maple to *check* your solutions, if applicable.

2 2 Part 1: Multiple Choice (1 mark each): Choose the best answer. 1. Which of the following expressions describe the set {2, 3, 4, 5}? a. {x Z 2 < x < 5} b. {x Z 2 x 5} c. {x Z 1 < x < 6} d. {x Z 1 x 6} e. Both b and c f. Both a and d g. All of the above h. None of the above 2. Which of the following intervals describes the set {x R 3 x < 10}? a. ( 3, 10) b. [ 3, 10) c. [ 3, 10] d. ( 3, 10] 3. Choose the mathematical statement that agrees with the following statement: The distance from x to 3 is less than 2. a. x 3 < 2 b. x 3 < 2 c. 0 < x 3 < 2 d. x Choose the mathematical statement that agrees with the following statement: The distance from 1 to x is less than or equal to 1 and x 1. 2 a. 0 x b. 0 < x 1 < 1 2 c. 0 < x d. 1 2 x 3 2

3 5. Choose the mathematical statement that agrees with the following statement: The distance from x to y is less than or equal to the sum of the distance from x to z plus the distance from z to y. a. x + y x + z z + y b. x + y x + z + z + y c. x y < x z + z y d. x y x z + z y 6. Which of the following Rules for Inequalities is FALSE? a. If a < b, then a + c < b + c. b. If a < b and c < d, then a + c < b + d. c. If a < b, then ac < bc. d. If a < b and c < 0, then ac > bc. e. If 0 < a < b, then 1 a > 1 b. f. None of the above (all statements are true) 7. Solve 2x 5 = 3? a. x = 4 b. x = 4 c. x = 1 d. x = 4 or x = 1 e. x = 4 or x = 1 f. None of the above 8. Solve 3x + 2 4? a. {x x 2 or x 2 3 } b. (, 2] [ 2 3, ) c. {x x 2 or x 2} 3 d. (, 2 ] [2, ) 3 e. Both A and B f. Both B and C g. None of the above 3

4 4 9. Given a sequence {a n } which of the following statements is the proper definition for convergence. a. The sequence {a n } converges to L if for each ɛ > 0 and for every cutoff N N, we have for all n N. a n L < ɛ b. The sequence {a n } converges to L if for each ɛ > 0 there is a cutoff N N, such that for all n N. a n L < ɛ c. The sequence {a n } converges to L if there exists an ɛ > 0 and cutoff N N, such that for all n N. a n L < ɛ d. The sequence {a n } converges to L if there exists an ɛ > 0 such that for every cutoff N N, we have a n L < ɛ for all n N Compute the limit of the sequence { 5n n+7 }. a. 0 b. 5 7 c. 1 7 d. 5 e. Does not exist 11. Compute the limit of the sequence { 100n+7000 n 2 n 1 }. a. 0 b. 1 c. 100 d e. Does not exist 12. Compute the limit of the sequence {cos(nπ)}. (Hint: compute the first few terms of the sequence.) a. 0 b. 1 c. -1 d. 2π e. Does not exist

5 5 13. Compute the limit of the sequence { n}. a. 0 b. 1 c. d. 2 e. Does not exist 14. The table lists the values of the first 9 values of a sequence. Which of the following statements must be true? a. The sequence has a limit of b. The sequence does not have a limit. n a n n a n n a n c. A limit for this sequence can not be determined from this information. d. All of the terms in the sequence will be positive. 15. Which of the following mathematical statements means that a. a n 0 = b. a n 0 < c. a n d. a n 0 > e. none of the above a n = 1 n 3 approximates 0 with an error of less than ? 16. Suppose that we want a n = 1 to approximate 0 with an error of less than 1. How large must n n be? (I.e., how far out in the sequence must you go so that {a n } is within 1 of 0?) 10 9 a. n > 1 b. n > 10 c. n > 100 d. n > 1000

6 6 For each of the following sequences in the next four questions choose all answers that apply. There may be more than one correct choice for each question. 17. Consider the sequence a n = 1 2n+3. Then a n is: a. increasing b. decreasing c. not monotonic d. bounded 18. Consider the sequence a n = 2n 3 3n+4. Then a n is: a. increasing b. decreasing c. not monotonic d. bounded 19. Consider the sequence a n = cos(nπ/2). Then a n is: a. increasing b. decreasing c. not monotonic d. bounded 20. Consider the sequence a n = n + 1 n. Then a n is: a. increasing b. decreasing c. not monotonic d. bounded 21. What is the sum of the series n=1 3? 10 n a b. 1 3 c d. 7 10

7 22. A drug is administered into the body. At the end of each hour, the amount of drug present is half what it was at the end of the previous hour. What fraction of the original amount of the drug is present at the end of n hours? a. ( 1 2 )n b. 1 2n c. 1 ( 1 2 )n d n 7 Part 2: True and False (1 mark each) In each of the following, indicate whether the statement is true (T) or false (F). 1. x a is the distance from x to a. 2. x = x 3. A sequence {a n } can have more than one limit. 4. The sequence {1, 1, 1, 1,...} has two limits. 5. If S R, then the number M is an upper bound of S if x M for every x S. 6. Let S = [0, 1). Then M = 1000 is an upper bound for S. 7. Let S = [0, 1). Then M = 1 is an upper bound for S. 8. Let S = [0, 1). Then M = 1 is the least upper bound for S. 9. If S is a non-empty set of real numbers, then S must have a least upper bound. 10. If {a n } is an increasing and bounded sequence, then {a n } must converge to its least upper bound. 11. If {a n } is an increasing and unbounded sequence, then {a n } diverges to. 12. It is possible to complete infinitely many tasks in a finite amount of time. 13. Even when we know that a series converges, most of the time it is difficult or even impossible to find its sum. 14. A geometric series converges when the absolute value of its common ratio is less than The harmonic series diverges. 16. If lim n a n = 0, then by the Divergence Test, n=1 a n converges.

8 8 Part 3: Calculation You must show enough work to justify your solution. 1) [9 marks total, 3 marks each] For each of the following sequences indicate whether it converges or diverges. If it converges find the limit. You must justify your answer. i) a n = 4n2 3n 2 n+1 ii) a n = n3 +4n n 5 +2n 2 iii) a n = n5 +2n+1 2n 3 +n+4

9 2) [4 marks] Use the Squeeze Theorem to determine the limit of the sequence { cos(3n) } if it exists. n 2 9

10 10 3) [8 marks total, 4 marks each] Let a n = 8n2 +4. Then lim a 4n 2 +1 n = 2. n i) Show that 8n2 +4 4n = 2 4n 2 +1 ( ) and use ( ) to show that if n 50, then 8n n < iii) Find the smallest cutoff N so that if n N, then 8n n <

11 11 4) [9 marks total] Calculate Let {a n } be the recursively defined sequence given by a 1 = 8 and a n+1 = 1 4 (3a n + 22 ) a 3 n a) [4 Marks] Use the Arithmetic Rules to show that if {a n } converges, then lim n a n = b) [1 Mark] Using a calculator or any other computational tool to find 4 22 up to 8 decimal places. Solution: 4 22 = c) [2 Marks] Find the smallest index n so that a n agrees with your answer in b) above to 8 decimal places. Solution: n = d) [2 Marks] Suppose that we had instead chosen a 1 = 2. Find the smallest index n so that a n agrees with your answer in ii) above to 8 decimal places. Solution: n =

12 12 5) [8 marks total] Suppose that a n 0 and that lim n a n = L. We want to show that lim n an = L. There are two cases to consider: a) [2 Marks] Suppose that L = 0. Let ɛ > 0. We want to choose a cutoff N 0 that is large enough so that if n N 0, then a n 0 < 1. To do so we should choose N N so that if n N 0 (Choose 1) i) then 0 a n < ii) then 0 a n < ( 1 iii) then 0 a n < 100 ) My Choice is: b) [2 Marks] Suppose that L = 5. Let ɛ > 0. Note that a n 5 = a n 5 an + 5 a n 5 5. We want to choose a cutoff N 0 that is large enough so that if n N 0, then a n L < To do so we should choose N 0 N so that if n N 0 (Choose 1) i) then a n L < ii) then a n L < iii) then a n L < My Choice is: It then follows from your choice above that if n N 0, then a n 5 < c) [4 Marks] Let a n = Note that a n 2 n 5. By making use of what we learned in b) above find a cutoff N 0 so that if n N 0, n 5 <

13 6) [18 marks total, 3 marks each, 6 bonus] Convergence of Subsequences: Recall that given a sequence {a n } and an increasing sequence n 1 < n 2 < < n k < of natural numbers {a nk } is called a subsequence of {a n }. Note that the questions marked bonus are at a higher level of difficulty and you would not be expected to create a solution to these questions on an exam. a) (Bonus Question 3 marks) Show directly using the definition of convergence for a sequence that if a sequence {a n } converges with lim a n = L, then any subsequence {a nk } also converges and lim a nk = L. n k 13 b) Let {a n } be a sequence such that lim a 2k = L and lim a 2k 1 = M. Explain why if L M, k k the sequence {a n } diverges.

14 14 6) Continued... c) Show that the sequence {( 1) n+1 ( 3n+2 )} diverges. 4n+1 d) (Bonus Question 3 marks) Once again let {a n } be a sequence such that lim a 2k = L and k lim a 2k 1 = M. Show that if L = M, then the sequence {a n } converges with lim a n = L. k n

15 6) Continued... ( ) While a sequence {a n } may not be monotonic, it can be shown that every sequence {a n } has a monotonic subsequence {a nk }. e) Use the statement ( ) above to show that if a sequence {a n } is bounded, that is if there exists an M > 0 such that M a n M for every n N, then it must have a convergent subsequence {a nk }. (Note: The fact that bounded sequences always have a convergent subsequence is known as the Bolzano-Weierstrass Theorem.) 15 f) Give an example of a sequence of {a n } with no convergent subsequences.

16 16 7) [4 marks] Consider the geometric series sum. n=1 1. Explain why it converges and find the value of the 10 n

MATH 137 : Calculus 1 for Honours Mathematics. Online Assignment #2. Introduction to Sequences

1 MATH 137 : Calculus 1 for Honours Mathematics Online Assignment #2 Introduction to Sequences Due by 9:00 pm on WEDNESDAY, September 19, 2018 Instructions: Weight: 2% This assignment covers the topics