AP Calc BC Convergence Tests Name: Block: Seat:
|
|
|
- Theresa Chloe Griffith
- 6 years ago
- Views:
Transcription
1 AP Calc BC Convergence Tests Name: Block: Seat: n th Term Divergence Test n=k diverges if lim n a n 0 a n diverges if lim n a n does not exist 1. Determine the convergence n 1 n + 1 Geometric Series The series ar n n=0 converges only if 1 < r < 1 the sum is a 1 r 1. Determine the convergence ( 3 5 ) n if it converges, find the sum St. Francis High School AP Calc BC
2 Page p Series Test The series diverges if p 1 converges if p > 1 n=k 1 n p 1. Determine the convergence 3 n n Direct Comparison Test This test is used when a known series is bigger than the given series. If a n has no negative terms, and a ceiling function k b n converges, then k a n must also converge. If a n has no negative terms, and a floor function k b n diverges, then k a n must also diverge. 1. Use the direct Comparison Test to determine the convergence 1 n St. Francis High School Page 2 12 AP Calc BC
3 Page Use the direct Comparison Test to determine the convergence 1 3 n Use the direct Comparison Test to determine the convergence 1 n 1/2 + 2 St. Francis High School Page 3 12 AP Calc BC
4 Page Limit Comparison Test This is one the most useful tests for determining convergence. Suppose a n > 0 and b n > 0 for all n > N where N is a positive integer. If lim n a n b n = c, 0 < c <, then a n and b n behave the same. 2. Use the Limit Comparison Test to determine the convergence 1 2 n 1 If lim n a n b n = 0, and b n converges, then a n converges. If lim n a n b n =, and b n diverges, then a n diverges. 1. Use the Limit Comparison Test to determine the convergence 3n + 2 (n + 1) 2 St. Francis High School Page 4 12 AP Calc BC
5 Page Use the Limit Comparison Test to determine the convergence 2n + 1 n 3 2n 4. Use the Limit Comparison Test to determine the convergence ln n n 3/2 St. Francis High School Page 5 12 AP Calc BC
6 Page Alternating Series Test for Convergence The series ( 1) n+1 a n = a 1 a 2 + a 3 a Use the AST to determine to determine the convergence 3( 1) n n will converge if all three true: Each Term is positive The terms are eventually decreasing lim n a n = 0 1. Use the AST to determine to determine the convergence ( 1) n n St. Francis High School Page 6 12 AP Calc BC
7 Page Use the AST to determine to determine the convergence 10( 1) n n! 4. Use the AST to determine to determine the convergence ( 1) n 2 n e n 1 St. Francis High School Page 7 12 AP Calc BC
8 Page Root Test The next test for convergence or divergence series works especially well for series involving nth powers. Let a n be a series. n=k n If lim an = L exists, then n If L < 1, then the series converges 2. Use the Root test to determine the convergence 3 n e n If L > 1 then the series diverges If L = 1 then the test is inconclusive 1. Use the Root test to determine the convergence [ n 2 ] n + 1 2n St. Francis High School Page 8 12 AP Calc BC
9 Page Use the Root test to determine the convergence e 3n n n 4. Use the Root test to determine the convergence [ 1 ] n arctan n St. Francis High School Page 9 12 AP Calc BC
10 Page Ratio Test This test is used most ten on the AP Exam to determine convergence a power series. Let n=k a n be a series with positive terms and lim a n+1 n a n = L 2. Use the Ratio Test to determine the interval and radius convergence (the values x that will make this converge): n=0 nx n 10 n If L < 1, then the series converges If L > 1 then the series diverges If L = 1 then the test is inconclusive 1. Use the Ratio Test to determine if the following converges: 2 n 3 n + 1 n=0 St. Francis High School Page AP Calc BC
11 Page Use the Ratio Test to determine the interval and radius convergence (the values x that will make this converge): (x + 5) n n=0 2. Find the interval convergence for ( ) x ( 1) n+1 2n 2n After the Ratio Test find the values x that will make the limit less than 1. This interval values x that make it converge is the interval convergence. Be sure to check the end points the interval 1. Find the interval convergence for n=4 nx n 4 n (n 2 + 1) St. Francis High School Page AP Calc BC
12 Page Find the interval convergence for (x 2) n n=4 3n 4. Find the interval convergence for x n n n 3 n St. Francis High School Page AP Calc BC
Root test. Root test Consider the limit L = lim n a n, suppose it exists. L < 1. L > 1 (including L = ) L = 1 the test is inconclusive.
Root test Root test n Consider the limit L = lim n a n, suppose it exists. L < 1 a n is absolutely convergent (thus convergent); L > 1 (including L = ) a n is divergent L = 1 the test is inconclusive.
Convergence Tests. Academic Resource Center
Convergence Tests Academic Resource Center Series Given a sequence {a 0, a, a 2,, a n } The sum of the series, S n = A series is convergent if, as n gets larger and larger, S n goes to some finite number.
Chapter 9: Infinite Series Part 2
Name: Date: Period: AP Calc BC Mr. Mellina/Ms. Lombardi Chapter 9: Infinite Series Part 2 Topics: 9.5 Alternating Series Remainder 9.7 Taylor Polynomials and Approximations 9.8 Power Series 9.9 Representation
The infinite series is written using sigma notation as: lim u k. lim. better yet, we can say if the
Divergence and Integral Test With the previous content, we used the idea of forming a closed form for the n th partial sum and taking its limit to determine the SUM of the series (if it exists). *** It
Absolute Convergence and the Ratio & Root Tests
Absolute Convergence and the Ratio & Root Tests Math114 Department of Mathematics, University of Kentucky February 20, 2017 Math114 Lecture 15 1/ 12 ( 1) n 1 = 1 1 + 1 1 + 1 1 + Math114 Lecture 15 2/ 12
AP Calculus Chapter 9: Infinite Series
AP Calculus Chapter 9: Infinite Series 9. Sequences a, a 2, a 3, a 4, a 5,... Sequence: A function whose domain is the set of positive integers n = 2 3 4 a n = a a 2 a 3 a 4 terms of the sequence Begin
Power Series. Part 1. J. Gonzalez-Zugasti, University of Massachusetts - Lowell
Power Series Part 1 1 Power Series Suppose x is a variable and c k & a are constants. A power series about x = 0 is c k x k A power series about x = a is c k x a k a = center of the power series c k =
10.1 Sequences. A sequence is an ordered list of numbers: a 1, a 2, a 3,..., a n, a n+1,... Each of the numbers is called a term of the sequence.
10.1 Sequences A sequence is an ordered list of numbers: a 1, a 2, a 3,..., a n, a n+1,... Each of the numbers is called a term of the sequence. Notation: A sequence {a 1, a 2, a 3,...} can be denoted
Infinite Series - Section Can you add up an infinite number of values and get a finite sum? Yes! Here is a familiar example:
Infinite Series - Section 10.2 Can you add up an infinite number of values and get a finite sum? Yes! Here is a familiar example: 1 3 0. 3 0. 3 0. 03 0. 003 0. 0003 Ifa n is an infinite sequence, then
2 2 + x =
Lecture 30: Power series A Power Series is a series of the form c n = c 0 + c 1 x + c x + c 3 x 3 +... where x is a variable, the c n s are constants called the coefficients of the series. n = 1 + x +
Math Exam II Review
Math 114 - Exam II Review Peter A. Perry University of Kentucky March 6, 2017 Bill of Fare 1. It s All About Series 2. Convergence Tests I 3. Convergence Tests II 4. The Gold Standards (Geometric Series)
A sequence { a n } converges if a n = finite number. Otherwise, { a n }
9.1 Infinite Sequences Ex 1: Write the first four terms and determine if the sequence { a n } converges or diverges given a n =(2n) 1 /2n A sequence { a n } converges if a n = finite number. Otherwise,
Math 162 Review of Series
Math 62 Review of Series. Explain what is meant by f(x) dx. What analogy (analogies) exists between such an improper integral and an infinite series a n? An improper integral with infinite interval of
Review Sheet on Convergence of Series MATH 141H
Review Sheet on Convergence of Series MATH 4H Jonathan Rosenberg November 27, 2006 There are many tests for convergence of series, and frequently it can been confusing. How do you tell what test to use?
Review of Power Series
Review of Power Series MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Introduction In addition to the techniques we have studied so far, we may use power
10.1 Sequences. Example: A sequence is a function f(n) whose domain is a subset of the integers. Notation: *Note: n = 0 vs. n = 1.
10.1 Sequences Example: A sequence is a function f(n) whose domain is a subset of the integers. Notation: *Note: n = 0 vs. n = 1 Examples: EX1: Find a formula for the general term a n of the sequence,
(Infinite) Series Series a n = a 1 + a 2 + a a n +...
(Infinite) Series Series a n = a 1 + a 2 + a 3 +... + a n +... What does it mean to add infinitely many terms? The sequence of partial sums S 1, S 2, S 3, S 4,...,S n,...,where nx S n = a i = a 1 + a 2
Math 0230 Calculus 2 Lectures
Math 00 Calculus Lectures Chapter 8 Series Numeration of sections corresponds to the text James Stewart, Essential Calculus, Early Transcendentals, Second edition. Section 8. Sequences A sequence is a
Let s Get Series(ous)
Department of Mathematics, Computer Science, and Statistics Bloomsburg University Bloomsburg, Pennsylvania 785 Let s Get Series(ous) Summary Presenting infinite series can be (used to be) a tedious and
Chapter 11 - Sequences and Series
Calculus and Analytic Geometry II Chapter - Sequences and Series. Sequences Definition. A sequence is a list of numbers written in a definite order, We call a n the general term of the sequence. {a, a
Analysis II: Basic knowledge of real analysis: Part IV, Series
.... Analysis II: Basic knowledge of real analysis: Part IV, Series Kenichi Maruno Department of Mathematics, The University of Texas - Pan American March 1, 2011 K.Maruno (UT-Pan American) Analysis II
Chapter 8. Infinite Series
8.4 Series of Nonnegative Terms Chapter 8. Infinite Series 8.4 Series of Nonnegative Terms Note. Given a series we have two questions:. Does the series converge? 2. If it converges, what is its sum? Corollary
AP Calculus (BC) Chapter 9 Test No Calculator Section Name: Date: Period:
WORKSHEET: Series, Taylor Series AP Calculus (BC) Chapter 9 Test No Calculator Section Name: Date: Period: 1 Part I. Multiple-Choice Questions (5 points each; please circle the correct answer.) 1. The
Sequence. A list of numbers written in a definite order.
Sequence A list of numbers written in a definite order. Terms of a Sequence a n = 2 n 2 1, 2 2, 2 3, 2 4, 2 n, 2, 4, 8, 16, 2 n We are going to be mainly concerned with infinite sequences. This means we
8.5 Taylor Polynomials and Taylor Series
8.5. TAYLOR POLYNOMIALS AND TAYLOR SERIES 50 8.5 Taylor Polynomials and Taylor Series Motivating Questions In this section, we strive to understand the ideas generated by the following important questions:
Because of the special form of an alternating series, there is an simple way to determine that many such series converge:
Section.5 Absolute and Conditional Convergence Another special type of series that we will consider is an alternating series. A series is alternating if the sign of the terms alternates between positive
5.9 Representations of Functions as a Power Series
5.9 Representations of Functions as a Power Series Example 5.58. The following geometric series x n + x + x 2 + x 3 + x 4 +... will converge when < x
Math 106: Review for Final Exam, Part II - SOLUTIONS. (x x 0 ) 2 = !
Math 06: Review for Final Exam, Part II - SOLUTIONS. Use a second-degree Taylor polynomial to estimate 8. We choose f(x) x and x 0 7 because 7 is the perfect cube closest to 8. f(x) x /3 f(7) 3 f (x) 3
8.1 Sequences. Example: A sequence is a function f(n) whose domain is a subset of the integers. Notation: *Note: n = 0 vs. n = 1.
8. Sequences Example: A sequence is a function f(n) whose domain is a subset of the integers. Notation: *Note: n = 0 vs. n = Examples: 6. Find a formula for the general term a n of the sequence, assuming
MATH115. Infinite Series. Paolo Lorenzo Bautista. July 17, De La Salle University. PLBautista (DLSU) MATH115 July 17, / 43
MATH115 Infinite Series Paolo Lorenzo Bautista De La Salle University July 17, 2014 PLBautista (DLSU) MATH115 July 17, 2014 1 / 43 Infinite Series Definition If {u n } is a sequence and s n = u 1 + u 2
Math Test #3 Info and Review Exercises
Math 181 - Test #3 Info and Review Exercises Fall 2018, Prof. Beydler Test Info Date: Wednesday, November 28, 2018 Will cover sections 10.1-10.4, 11.1-11.7. You ll have the entire class to finish the test.
Power Series. Part 2 Differentiation & Integration; Multiplication of Power Series. J. Gonzalez-Zugasti, University of Massachusetts - Lowell
Power Series Part 2 Differentiation & Integration; Multiplication of Power Series 1 Theorem 1 If a n x n converges absolutely for x < R, then a n f x n converges absolutely for any continuous function
Taylor and Maclaurin Series. Approximating functions using Polynomials.
Taylor and Maclaurin Series Approximating functions using Polynomials. Approximating f x = e x near x = 0 In order to approximate the function f x = e x near x = 0, we can use the tangent line (The Linear
Final exam (practice) UCLA: Math 31B, Spring 2017
Instructor: Noah White Date: Final exam (practice) UCLA: Math 3B, Spring 207 This exam has 8 questions, for a total of 80 points. Please print your working and answers neatly. Write your solutions in the
Ma 530 Power Series II
Ma 530 Power Series II Please note that there is material on power series at Visual Calculus. Some of this material was used as part of the presentation of the topics that follow. Operations on Power Series
First In-Class Exam Solutions Math 410, Professor David Levermore Monday, 1 October 2018
First In-Class Exam Solutions Math 40, Professor David Levermore Monday, October 208. [0] Let {b k } k N be a sequence in R and let A be a subset of R. Write the negations of the following assertions.
11.4 The Comparison Tests. Copyright Cengage Learning. All rights reserved.
11.4 The Comparison Tests Copyright Cengage Learning. All rights reserved. The Comparison Tests In the comparison tests the idea is to compare a given series with a series that is known to be convergent
Lecture 32: Taylor Series and McLaurin series We saw last day that some functions are equal to a power series on part of their domain.
Lecture 32: Taylor Series and McLaurin series We saw last day that some functions are equal to a power series on part of their domain. For example f(x) = 1 1 x = 1 + x + x2 + x 3 + = ln(1 + x) = x x2 2
H2 MATHS SET D PAPER 1
H Maths Set D Paper H MATHS Exam papers with worked solutions SET D PAPER Compiled by THE MATHS CAFE P a g e b The curve y ax c x 3 points, and, H Maths Set D Paper has a stationary point at x 3. It also
Math Absolute Convergence, Ratio Test, Root Test
Math 114 - Absolute Convergence, Ratio Test, Root Test Peter A. Perry University of Kentucky February 20, 2017 Bill of Fare 1. Review and Recap 2. Dirichlet s Dilemma 3. Absolute Convergence 4. Ratio Test
Exam 3. Math Spring 2015 April 8, 2015 Name: } {{ } (from xkcd) Read all of the following information before starting the exam:
Exam 3 Math 2 - Spring 205 April 8, 205 Name: } {{ } by writing my name I pledge to abide by the Emory College Honor Code (from xkcd) Read all of the following information before starting the exam: For
Math 113: Quiz 6 Solutions, Fall 2015 Chapter 9
Math 3: Quiz 6 Solutions, Fall 05 Chapter 9 Keep in mind that more than one test will wor for a given problem. I chose one that wored. In addition, the statement lim a LR b means that L Hôpital s rule
Without fully opening the exam, check that you have pages 1 through 12.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through 2. Show all your work on the standard response
1. Pace yourself. Make sure you write something on every problem to get partial credit. 2. If you need more space, use the back of the previous page.
***THIS TIME I DECIDED TO WRITE A LOT OF EXTRA PROBLEMS TO GIVE MORE PRACTICE. The actual midterm will have about 6 problems. If you want to practice something with approximately the same length as the
Have a Safe Winter Break
SI: Math 122 Final December 8, 2015 EF: Name 1-2 /20 3-4 /20 5-6 /20 7-8 /20 9-10 /20 11-12 /20 13-14 /20 15-16 /20 17-18 /20 19-20 /20 Directions: Total / 200 1. No books, notes or Keshara in any word
Final exam (practice) UCLA: Math 31B, Spring 2017
Instructor: Noah White Date: Final exam (practice) UCLA: Math 31B, Spring 2017 This exam has 8 questions, for a total of 80 points. Please print your working and answers neatly. Write your solutions in
Introduction to Series and Sequences Math 121 Calculus II Spring 2015
Introduction to Series and Sequences Math Calculus II Spring 05 The goal. The main purpose of our study of series and sequences is to understand power series. A power series is like a polynomial of infinite
The integral test and estimates of sums
The integral test Suppose f is a continuous, positive, decreasing function on [, ) and let a n = f (n). Then the series n= a n is convergent if and only if the improper integral f (x)dx is convergent.
Math 142, Final Exam. 12/7/10.
Math 4, Final Exam. /7/0. No notes, calculator, or text. There are 00 points total. Partial credit may be given. Write your full name in the upper right corner of page. Number the pages in the upper right
Solutions to Math 1b Midterm II
Solutions to Math b Midterm II Tuesday, pril 8, 006. (6 points) Suppose that the power series a n(x + ) n converges if x = 7 and diverges if x = 7. ecide which of the following series must converge, must
SERIES REVIEW SHEET, SECTIONS 11.1 TO 11.5 OF OZ
SERIES REVIEW SHEET, SECTIONS 11.1 TO 11.5 OF OZ Fill in the blanks and give the indicated examples, including reasons. Don t simply fill in the blanks and give the examples. Take this opportunity to really
Multiple Choice. (c) 1 (d)
Multiple Choice.(5 pts.) Find the sum of the geometric series n=0 ( ) n. (c) (d).(5 pts.) Find the 5 th Maclaurin polynomial for the function f(x) = sin x. (Recall that Maclaurin polynomial is another
Power series and Taylor series
Power series and Taylor series D. DeTurck University of Pennsylvania March 29, 2018 D. DeTurck Math 104 002 2018A: Series 1 / 42 Series First... a review of what we have done so far: 1 We examined series
Math 122 Test 3. April 17, 2018
SI: Math Test 3 April 7, 08 EF: 3 4 5 6 7 8 9 0 Total Name Directions:. No books, notes or April showers. You may use a calculator to do routine arithmetic computations. You may not use your calculator
Name. Instructor K. Pernell 1. Berkeley City College Due: HW 4 - Chapter 11 - Infinite Sequences and Series. Write the first four terms of {an}.
Berkeley City College Due: HW 4 - Chapter 11 - Infinite Sequences and Series Name Write the first four terms of {an}. 1) an = (-1)n n 2) an = n + 1 3n - 1 3) an = sin n! 3 Determine whether the sequence
Analysis II: Basic knowledge of real analysis: Part V, Power Series, Differentiation, and Taylor Series
.... Analysis II: Basic knowledge of real analysis: Part V, Power Series, Differentiation, and Taylor Series Kenichi Maruno Department of Mathematics, The University of Texas - Pan American March 4, 20
Series. Definition. a 1 + a 2 + a 3 + is called an infinite series or just series. Denoted by. n=1
Definition a 1 + a 2 + a 3 + is called an infinite series or just series. Denoted by a n, or a n. Chapter 11: Sequences and, Section 11.2 24 / 40 Given a series a n. The partial sum is the sum of the first
MTH 133 Exam 2 November 16th, Without fully opening the exam, check that you have pages 1 through 12.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through 2. Show all your work on the standard response
Infinite Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Infinite Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Background Consider the repeating decimal form of 2/3. 2 3 = 0.666666 = 0.6 + 0.06 + 0.006 + 0.0006 + = 6(0.1)
Math 122 Test 3. April 15, 2014
SI: Math 1 Test 3 April 15, 014 EF: 1 3 4 5 6 7 8 Total Name Directions: 1. No books, notes or 6 year olds with ear infections. You may use a calculator to do routine arithmetic computations. You may not
Introduction and Review of Power Series
Introduction and Review of Power Series Definition: A power series in powers of x a is an infinite series of the form c n (x a) n = c 0 + c 1 (x a) + c 2 (x a) 2 +...+c n (x a) n +... If a = 0, this is
Math 113 (Calculus 2) Exam 4
Math 3 (Calculus ) Exam 4 November 0 November, 009 Sections 0, 3 7 Name Student ID Section Instructor In some cases a series may be seen to converge or diverge for more than one reason. For such problems
Math 162: Calculus IIA
Math 62: Calculus IIA Final Exam ANSWERS December 9, 26 Part A. (5 points) Evaluate the integral x 4 x 2 dx Substitute x 2 cos θ: x 8 cos dx θ ( 2 sin θ) dθ 4 x 2 2 sin θ 8 cos θ dθ 8 cos 2 θ cos θ dθ
Homework Problem Answers
Homework Problem Answers Integration by Parts. (x + ln(x + x. 5x tan 9x 5 ln sec 9x 9 8 (. 55 π π + 6 ln 4. 9 ln 9 (ln 6 8 8 5. (6 + 56 0/ 6. 6 x sin x +6cos x. ( + x e x 8. 4/e 9. 5 x [sin(ln x cos(ln
Without fully opening the exam, check that you have pages 1 through 12.
MTH 33 Exam 2 November 4th, 208 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through 2. Show
Taylor and Maclaurin Series. Copyright Cengage Learning. All rights reserved.
11.10 Taylor and Maclaurin Series Copyright Cengage Learning. All rights reserved. We start by supposing that f is any function that can be represented by a power series f(x)= c 0 +c 1 (x a)+c 2 (x a)
TAYLOR AND MACLAURIN SERIES
TAYLOR AND MACLAURIN SERIES. Introduction Last time, we were able to represent a certain restricted class of functions as power series. This leads us to the question: can we represent more general functions
AP Calculus Testbank (Chapter 9) (Mr. Surowski)
AP Calculus Testbank (Chapter 9) (Mr. Surowski) Part I. Multiple-Choice Questions n 1 1. The series will converge, provided that n 1+p + n + 1 (A) p > 1 (B) p > 2 (C) p >.5 (D) p 0 2. The series
MTH 133 Solutions to Exam 2 April 19, Without fully opening the exam, check that you have pages 1 through 12.
MTH 33 Solutions to Exam 2 April 9, 207 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through
The Comparison Test & Limit Comparison Test
The Comparison Test & Limit Comparison Test Math4 Department of Mathematics, University of Kentucky February 5, 207 Math4 Lecture 3 / 3 Summary of (some of) what we have learned about series... Math4 Lecture
Convergence Tests. Theorem. (the divergence test)., then the series u k diverges. k k. (b) If lim u = 0, then the series u may either converge
Convergence Tests We are now interested in developing tests to tell whether or not a series converges, without having to guess at its sum. The first such test is entirely a negative result. Theorem. (the
MTH 133 Solutions to Exam 2 November 15, Without fully opening the exam, check that you have pages 1 through 13.
MTH 33 Solutions to Exam 2 November 5, 207 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through
Without fully opening the exam, check that you have pages 1 through 13.
MTH 33 Solutions to Exam November th, 08 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through
Math 115 HW #5 Solutions
Math 5 HW #5 Solutions From 29 4 Find the power series representation for the function and determine the interval of convergence Answer: Using the geometric series formula, f(x) = 3 x 4 3 x 4 = 3(x 4 )
Last/Family Name First/Given Name Seat #
Math 2, Fall 27 Schaeffer/Kemeny Final Exam (December th, 27) Last/Family Name First/Given Name Seat # Failure to follow the instructions below will constitute a breach of the Stanford Honor Code: You
Math 113 Winter 2005 Departmental Final Exam
Name Student Number Section Number Instructor Math Winter 2005 Departmental Final Exam Instructions: The time limit is hours. Problem consists of short answer questions. Problems 2 through are multiple
Assignment 16 Assigned Weds Oct 11
Assignment 6 Assigned Weds Oct Section 8, Problem 3 a, a 3, a 3 5, a 4 7 Section 8, Problem 4 a, a 3, a 3, a 4 3 Section 8, Problem 9 a, a, a 3, a 4 4, a 5 8, a 6 6, a 7 3, a 8 64, a 9 8, a 0 56 Section
Chapter 1. Pigeonhole Principle
Chapter 1. Pigeonhole Principle Prof. Tesler Math 184A Fall 2017 Prof. Tesler Ch. 1. Pigeonhole Principle Math 184A / Fall 2017 1 / 16 Pigeonhole principle https://commons.wikimedia.org/wiki/file:toomanypigeons.jpg
Series Handout A. 1. Determine which of the following sums are geometric. If the sum is geometric, express the sum in closed form.
Series Handout A. Determine which of the following sums are geometric. If the sum is geometric, exress the sum in closed form. 70 a) k= ( k ) b) 50 k= ( k )2 c) 60 k= ( k )k d) 60 k= (.0)k/3 2. Find the
Preliminary check: are the terms that we are adding up go to zero or not? If not, proceed! If the terms a n are going to zero, pick another test.
Throughout these templates, let series. be a series. We hope to determine the convergence of this Divergence Test: If lim is not zero or does not exist, then the series diverges. Preliminary check: are
1. (25 points) Consider the region bounded by the curves x 2 = y 3 and y = 1. (a) Sketch both curves and shade in the region. x 2 = y 3.
Test Solutions. (5 points) Consider the region bounded by the curves x = y 3 and y =. (a) Sketch both curves and shade in the region. x = y 3 y = (b) Find the area of the region above. Solution: Observing
Spring 2017 Exam 2 NAME: PIN:
MARK BOX problem points 0 10 HAND IN PART 1 10 2 10 3-10 40=8x5 11 10 12 10 NAME: PIN: 13 10 % 100 INSTRUCTIONS This exam comes in two parts. (1) HAND-IN PART. Hand-in only this part. (2) NOT TO HAND-IN
Series. richard/math230 These notes are taken from Calculus Vol I, by Tom M. Apostol,
Series Professor Richard Blecksmith richard@math.niu.edu Dept. of Mathematical Sciences Northern Illinois University http://math.niu.edu/ richard/math230 These notes are taken from Calculus Vol I, by Tom
Functions. Given a function f: A B:
Functions Given a function f: A B: We say f maps A to B or f is a mapping from A to B. A is called the domain of f. B is called the codomain of f. If f(a) = b, then b is called the image of a under f.
Math 1b Sequences and series summary
Math b Sequences and series summary December 22, 2005 Sequences (Stewart p. 557) Notations for a sequence: or a, a 2, a 3,..., a n,... {a n }. The numbers a n are called the terms of the sequence.. Limit
Math Review for Exam Answer each of the following questions as either True or False. Circle the correct answer.
Math 22 - Review for Exam 3. Answer each of the following questions as either True or False. Circle the correct answer. (a) True/False: If a n > 0 and a n 0, the series a n converges. Soln: False: Let
Without fully opening the exam, check that you have pages 1 through 12.
MTH 33 Exam 2 April th, 208 Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through 2. Show all
Spring 2018 Exam 2 MARK BOX HAND IN PART NAME: PIN: INSTRUCTIONS
Spring 208 Exam 2 problem MARK BOX points HAND IN PART 0 0 4 2-5 56=4x4 6 0 7 0 NAME: PIN: % 00 INSTRUCTIONS This exam comes in two parts () HAND IN PART Hand in only this part (2) STATEMENT OF MULTIPLE
The Integral Test. P. Sam Johnson. September 29, P. Sam Johnson (NIT Karnataka) The Integral Test September 29, / 39
The Integral Test P. Sam Johnson September 29, 207 P. Sam Johnson (NIT Karnataka) The Integral Test September 29, 207 / 39 Overview Given a series a n, we have two questions:. Does the series converge?
You can learn more about the services offered by the teaching center by visiting
MAC 232 Exam 3 Review Spring 209 This review, produced by the Broward Teaching Center, contains a collection of questions which are representative of the type you may encounter on the exam. Other resources
MA EXAM 3 INSTRUCTIONS VERSION 01 April 18, Section # and recitation time
MA 16600 EXAM 3 INSTRUCTIONS VERSION 01 April 18, 2018 Your name Student ID # Your TA s name Section # and recitation time 1. You must use a #2 pencil on the scantron sheet (answer sheet). 2. Check that
Ratio Test Recall that every convergent series X a k either
Ratio Test Recall that every convergent series X a either X converges absolutely a converges, thus so does X a,or X converges conditionally a converges, but X a does not We will loo at two tests (Rato
Representation of Functions as Power Series.
MATH 0 - A - Spring 009 Representation of Functions as Power Series. Our starting point in this section is the geometric series: x n = + x + x + x 3 + We know this series converges if and only if x
MA 114 Worksheet # 1: Improper Integrals
MA 4 Worksheet # : Improper Integrals. For each of the following, determine if the integral is proper or improper. If it is improper, explain why. Do not evaluate any of the integrals. (c) 2 0 2 2 x x
CHAPTER 2 INFINITE SUMS (SERIES) Lecture Notes PART 1
CHAPTER 2 INFINITE SUMS (SERIES) Lecture Notes PART We extend now the notion of a finite sum Σ n k= a k to an INFINITE SUM which we write as Σ n= a n as follows. For a given a sequence {a n } n N {0},
C.7. Numerical series. Pag. 147 Proof of the converging criteria for series. Theorem 5.29 (Comparison test) Let a k and b k be positive-term series
C.7 Numerical series Pag. 147 Proof of the converging criteria for series Theorem 5.29 (Comparison test) Let and be positive-term series such that 0, for any k 0. i) If the series converges, then also
Section Taylor and Maclaurin Series
Section.0 Taylor and Maclaurin Series Ruipeng Shen Feb 5 Taylor and Maclaurin Series Main Goal: How to find a power series representation for a smooth function us assume that a smooth function has a power
3! + 4! + Binomial series: if α is a nonnegative integer, the series terminates. Otherwise, the series converges if x < 1 but diverges if x > 1.
Page 1 Name: ID: Section: This exam has 16 questions: 14 multiple choice questions worth 5 points each. hand graded questions worth 15 points each. Important: No graphing calculators! Any non-graphing
Section 9.8. First let s get some practice with determining the interval of convergence of power series.
First let s get some practice with determining the interval of convergence of power series. First let s get some practice with determining the interval of convergence of power series. Example (1) Determine
Testing Series with Mixed Terms
Testing Series with Mixed Terms Philippe B. Laval KSU Today Philippe B. Laval (KSU) Series with Mixed Terms Today 1 / 17 Outline 1 Introduction 2 Absolute v.s. Conditional Convergence 3 Alternating Series