Fourier Analysis. Lecture 1: Sequences and Series. v1 by G. M. Chávez-Campos. Updated: 2016/06/27. Instituto Tecnológico de Morelia
|
|
- Coral Williamson
- 5 years ago
- Views:
Transcription
1 Fourier Analysis Lecture 1: Sequences and Series v1 by G. M. Chávez-Campos Updated: 2016/06/27 Instituto Tecnológico de Morelia
2 Course information
3 Course information La asignatura le aporta al estudiante herramientas anaĺıticas, relacionadas con matemáticas avanzadas, que le permiten resolver problemas de ecuaciones y circuitos relacionados con la electrónica analógica y digital. Didactics El estudiante debe realizar actividades que le permitan desarrollar las competencias necesarias para comprender el uso de series, sucesiones y transformadas de Fourier utilizadas en la solución de problemas en electrónica analógica y digital.
4 Course Target El alumno debe comprender el uso de series, sucesiones y transformadas de Fourier utilizadas en la solución de problemas en electrónica analógica-digital.
5 Course topics Lecture 1: Sequences, series and Fourier series. Lecture 2: Fourier Transform. Lecture 3: Inverse continuos Fourier transform. Lecture 4: Discrete Fourier transform. Lecture 5: Inverse discrete Fourier transform. Lecture 6: Fast Fourier Transform.
6 Overview 1. Course information 2. Sequences 2.1 Introduction: practical cases 2.2 Type of sequences 3. Series 3.1 Finite and infinite series 3.2 Convergency 3.3 The kth term theorem 3.4 Special series Arithmetic series Harmonic series Geometric series
7 Sequences
8 Practical situations Try to solve the next practical cases: Case 1: Consider that you are throwing an object from the first floor of a building, the object falls 16ft vertical distance; from second floor, the object falls 48ft; third floor, 80ft; and so on. How many distance will falls the object in floor 4, 5 and 6? Case 2: A bacterium is able to reproduce itself by double every 15 minutes. How many bacteria will be after 2 hours?
9 Sequences Definition A sequence is a number of people or things of a similar kind following one after the other. Also the sequence could be a group of real numbers. This sequence is said to be finite because there is a first and last number. If the set of numbers which defines a sequence does no have both a first and last number, the sequence is said to be infinite.
10 Sequences cont. Finite 1,2,4,8-24,-20,-16,...,0
11 Sequences cont. Finite 1,2,4,8-24,-20,-16,...,0 Infinite 1,3,6,... 0,5,10,15,20,...
12 Recursive Definition A recursive succession commonly offers the next information: First term or first terms, it is specified some way to obtain next terms, using the first terms
13 Hands on... The student may calculate the first five terms for the succession a 1 = 5 and a n = a n for n {2, 3, 4...}: a 1 = 5 a 2 a 3 a 4 a 5
14 Exercise 2 The student may calculate the first five terms for the succession a 1 = 4 and a n = 1 2 a n 1 for n 2: a 1 = 4 a 2 a 3 a 4 a 5
15 Series
16 Series Definition If {U n } is a sequence and S n = n U i = U 1 + U 2 + U U n i=1 then the sequence {S n } is called an infinite series. The numbers U 1, U 2, U 3,... are called the terms.
17 Series: Finite and infinite series S n = U n = U 1 + U 2 + U U n + (1) n=1 S k = k U i = U 1 + U 2 + U U k (2) i=1 where (1) is an infinity series and (2) is kth partial term of a given series. sy
18 Exercise 3: Given the infinity series: n=1 1 n(n + 1) find the first four elements of the sequence of partial sums {s n }, and find a formula for s n in terms of n.
19 Convergent series Definition Let n=1 U n be a given infinite series, and let {S n } be the sequence of partial sums defining this infinity series. Then if lim S n exists and is equal to S, we say that the given series is convergent and that S is the sum of the given infinity series. If lim S n does no exist, the series is said to be divergent and the series does not have a sum.
20 Exercise 4: Given the infinity series: + n=1 1 n(n + 1)
21 Exercise 4: Given the infinity series: + n=1 lim S n = n + 1 n(n + 1) lim n + n n + 1
22 Exercise 4: Given the infinity series: + n=1 lim S n = n + 1 n(n + 1) n=1 lim n + n n + 1 n n + 1
23 The k th term theorem Theorem If the infinity series + n=1 U n is convergent If the infinity series + n=1 U n is divergent lim U n = 0 (3) n + lim U n 0 (4) n +
24 Exercise 5: + n=1 + n=1 n n 2 (5) 3 ( 1) n+1 (6)
25 Harmonic series Note that the Theorem (3) not always is true. In other words, it is possible to have a divergent series for which lim n + U n = 0. An example of such a series is the one known as the harmonic, which is + n=1 1 n = n (7)
26 Does the harmonic series converges? Theorem Let {S n } be the sequence of partial sums for a given convergent series + n=1 U n. Then for any ɛ > 0 there exist a number N such that S R S T < ɛ, whenever R > N and T > N
27 Does the harmonic series converges? Theorem Let {S n } be the sequence of partial sums for a given convergent series + n=1 U n. Then for any ɛ > 0 there exist a number N such that S R S T < ɛ, whenever R > N and T > N Copyright by [2]
28 Geometric series The geometric series is a sires of the form + n=1 Lets try to understand... ar n 1 = a + ar + ar ar n 1 + (8)
29 Geometric series The geometric series is a sires of the form + n=1 Lets try to understand... ar n 1 = a + ar + ar ar n 1 + (8) S n = a(1 r n ) 1 r (9)
30 Geometric series convergency Theorem The geometric series converges to the sum a/(1 r) if r < 1, and the geometric series diverges if r >= 1
31 Exercise 6: Express the decimal number as a common fraction.
32 Two series theorem Theorem If + n=1 a n and + n=1 b ntwo infinite series, differing only in their first m terms, then either both series converge or both series diverge.
33 Two series theorem Theorem If + n=1 a n and + n=1 b ntwo infinite series, differing only in their first m terms, then either both series converge or both series diverge. Theorem If the series + n=1 a n is convergent and its sum is S, then the series + n=1 ca n is also convergent and its sum is c S If the series + n=1 a n is divergent, then the series + n=1 ca n is also divergent
34 Exercise 7: Determine whether the infinity series is convergent or divergent + n=1 1 4n
35 The integral test The theorem known as the integral test makes use of the theory of improper integrals to test an infinite series of positive terms for convergence.
36 The integral test The theorem known as the integral test makes use of the theory of improper integrals to test an infinite series of positive terms for convergence. Theorem Let f be a function which is continuous, decreasing, and positive valued for all x >= 1, then the infinite series + n=1 is convergent if the improper integral f (n) = f (1) + f (2) + + f (n) f (x)dx exists, and is divergent if the improper integral increase without bound.
37 Theorem Let a n be an infinite series of nonzero terms and let L to be calculated by (10) lim a n+1 n + a n = L (10) thus:
38 Theorem Let a n be an infinite series of nonzero terms and let L to be calculated by (10) lim a n+1 n + a n = L (10) thus: If L < 1, the series is absolutely convergent.
39 Theorem Let a n be an infinite series of nonzero terms and let L to be calculated by (10) lim a n+1 n + a n = L (10) thus: If L < 1, the series is absolutely convergent. If L > 1, or L, the series is divergent.
40 Theorem Let a n be an infinite series of nonzero terms and let L to be calculated by (10) lim a n+1 n + a n = L (10) thus: If L < 1, the series is absolutely convergent. If L > 1, or L, the series is divergent. If L = 1, the series may be absolutely convergent, conditionally convergent, or divergent.
41 Theorem Let a n be an infinite series of nonzero terms and let L to be calculated by (10) lim a n+1 n + a n = L (10) thus: If L < 1, the series is absolutely convergent. If L > 1, or L, the series is divergent. If L = 1, the series may be absolutely convergent, conditionally convergent, or divergent.
42 Exercise 8: Determine whether the following series are absolutely convergent, conditionally convergent, or divergent. ( 1) n 3n n! (11) 3 n n 2 (12) n=1 n=1
43 Power series
44 Power series Theorem Let x be a variable. A power series in x is a series of the form [2] a n x n = a 0 + a 1 x + a 2 x a n x n + (13) n=0 where each a n is a real number.
45 Exercise 9: Find all values of x for which the following power series is absolutely convergent x x n 5 n x n + (14) ! x + 1 2! x n! x n + (15)
46 Power series convergency theorem Theorem If a n x n is a power series, then precisely one of the following is true The series converges only if x = 0, The series is absolutely convergent for all x, There is a positive number r such that the series is absolutely convergent if x < r and divergent if x > r
47 convergency radius
48 Power series x c Theorem Let c be a real number and x a variable. A power series in x c is a series of the form + n=0 a n (x c) n = a 0 + a 1 (x c) + a 2 (x c) a n (x c) n + where each a n is a real number. (16)
49 Power series convergency theorem Theorem If a n (x c) n is a power series, then precisely one of the following is true The series converges only if x c = 0, The series is absolutely convergent for all x, There is a positive number r such that the series is absolutely convergent if x c < r and divergent if x c > r
50 convergency radius
51 Exercise 10: Find the interval of convergence of (x 3) (x 3)2 + (17)
52 Power series representations of function A power series a n x n may be used to define a function of f whose domain is the interval of convergence of the series. Specifically, for each x in this interval we let f (x) equal the sum of the series, that is f (x) = a 0 + a 1 x + a 2 x a n x n + (18) If a function f is defined in this way we say that a n x n is a power series representative for f (x). This allows us to find values in a new way. Specifically, if c is the interval of convergence, thus f (c) can be found or approximated be the series...
53 Exercise 11: Find a function f that is represented by the power series ( 1) n x n (19) n=0
54 Theorem Suppose a power series a n x n has a nonzero radius of convergence r and let the function f be defined by f (x) = a n x n (20) n=0 for every x in the interval of convergence. If r < x < r, then: f (x) = D x (a n x n ) = n=0 na n x (n 1) (21) n=1 = a 1 + 2a 2 x + 3a 3 x na n x (n 1) +
55 Integral Theorem x 0 f (t)dt = n=0 x 0 (a n t n )dt = n=0 a n n + 1 x n+1 (22) = a 0 x a 1x a 2x n + 1 a nx n+1 +
56 Tylor and Maclaurin Series
57 Tylor and Maclaurin Series Suppose a function f is represented by a power series in x c, such that f (x) = + n=0 a n (x c) n = a 0 +a 1 (x c)+a 2 (x c) 2 +a 3 (x c) 3 + where the domain of f is an open interval containing c (23)
58 f (x) = na n (x c) n 1 (24) n=1 = a 1 + 2a 2 (x c) + 3a 3 (x c) 2 + 4a 4 (x c) 3 + f (x) = n(n 1)a n (x c) n 2 (25) n=2 = 2a 2 + (3 2)a 3 (x c) + (4 3)a 4 (x c) 2 + f (x) = n(n 1)(n 2)a n (x c) n 3 (26) n=3 = (3 2)a 3 + (4 3 2)a 4 (x c) +
59 and, for every positive integer k, f (k) (x) = n(n 1) (n k + 1)a n (x c) n k (27) n=k Moreover, each series obtained by differentiation has the same radius of convergence as the original series. Substituting c for x in each of these series representation, we obtain f (c) = a 0 (28) f (c) = a 1 (29) f (c) = 2a 2 (30) f (c) = (3 2)a 3 (31) f (n) (c) = n!a n a n = f (n) (c) n! (32)
60 Taylor series Theorem If f is a function and f (x) = a n (x c) n (33) n=0 for all x in an open interval containing c, then f (x) = f (c) + f (c)(x c) + f (c) 2! (x c) f (n) (c) n! (x c) n (34)
61 Maclaurin Corollary If f is a function and f (x) = a n x n for all x in an open interval ( r, r), then f (x) = f (0) + f (0)x + f (0) 2! x f (n) (0) x n + (35) n!
62 References Louis Leithold The calculus with analytic geometry Harper & Row, Publishers, 1976 Earl W. Swokowski Calculus with analytic geometry Prindle, Weber & Schmidt, 1979 Théorie analytique de la chaleur Jean-Baptiste Joseph Fourier 1822 Sucesiones y Series Roberto O. Rivera Rodríguez 2015
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
More informationReview 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
More informationMath 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
More information10.1 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,
More informationAP 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
More informationAP 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
More information(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
More information12.1 Arithmetic Progression Geometric Progression General things about sequences
ENGR11 Engineering Mathematics Lecture Notes SMS, Victoria University of Wellington Week Five. 1.1 Arithmetic Progression An arithmetic progression is a sequence where each term is found by adding a fixed
More informationSeries. 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
More informationSection 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
More informationModule 9 : Infinite Series, Tests of Convergence, Absolute and Conditional Convergence, Taylor and Maclaurin Series
Module 9 : Infinite Series, Tests of Convergence, Absolute and Conditional Convergence, Taylor and Maclaurin Series Lecture 27 : Series of functions [Section 271] Objectives In this section you will learn
More information2 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 +
More informationAnalysis 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
More informationLet 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
More informationCALCULUS JIA-MING (FRANK) LIOU
CALCULUS JIA-MING (FRANK) LIOU Abstract. Contents. Power Series.. Polynomials and Formal Power Series.2. Radius of Convergence 2.3. Derivative and Antiderivative of Power Series 4.4. Power Series Expansion
More informationA 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,
More informationMath 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
More informationTaylor 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)
More informationTesting 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
More informationIntroduction 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
More informationPower 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 =
More informationPower 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
More informationSequences and Series, Induction. Review
Sequences and Series, Induction Review 1 Topics Arithmetic Sequences Arithmetic Series Geometric Sequences Geometric Series Factorial Notation Sigma Notation Binomial Theorem Mathematical Induction 2 Arithmetic
More informationMath 141: Lecture 19
Math 141: Lecture 19 Convergence of infinite series Bob Hough November 16, 2016 Bob Hough Math 141: Lecture 19 November 16, 2016 1 / 44 Series of positive terms Recall that, given a sequence {a n } n=1,
More informationLECTURE 10: REVIEW OF POWER SERIES. 1. Motivation
LECTURE 10: REVIEW OF POWER SERIES By definition, a power series centered at x 0 is a series of the form where a 0, a 1,... and x 0 are constants. For convenience, we shall mostly be concerned with the
More informationInfinite 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)
More information10.1 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
More information3.4 Introduction to power series
3.4 Introduction to power series Definition 3.4.. A polynomial in the variable x is an expression of the form n a i x i = a 0 + a x + a 2 x 2 + + a n x n + a n x n i=0 or a n x n + a n x n + + a 2 x 2
More informationHomework 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
More informationINTRODUCTION TO REAL ANALYSIS II MATH 4332 BLECHER NOTES
INTRODUCTION TO REAL ANALYSIS II MATH 4332 BLECHER NOTES You will be expected to reread and digest these typed notes after class, line by line, trying to follow why the line is true, for example how it
More informationInfinite 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
More informationAP Calc BC Convergence Tests Name: Block: Seat:
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
More informationSequences: Limit Theorems
Sequences: Limit Theorems Limit Theorems Philippe B. Laval KSU Today Philippe B. Laval (KSU) Limit Theorems Today 1 / 20 Introduction These limit theorems fall in two categories. 1 The first category deals
More informationMath 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
More informationAdvanced Calculus Math 127B, Winter 2005 Solutions: Final. nx2 1 + n 2 x, g n(x) = n2 x
. Define f n, g n : [, ] R by f n (x) = Advanced Calculus Math 27B, Winter 25 Solutions: Final nx2 + n 2 x, g n(x) = n2 x 2 + n 2 x. 2 Show that the sequences (f n ), (g n ) converge pointwise on [, ],
More informationCompletion Date: Monday February 11, 2008
MATH 4 (R) Winter 8 Intermediate Calculus I Solutions to Problem Set #4 Completion Date: Monday February, 8 Department of Mathematical and Statistical Sciences University of Alberta Question. [Sec..9,
More informationSequences and Series
CHAPTER Sequences and Series.. Convergence of Sequences.. Sequences Definition. Suppose that fa n g n= is a sequence. We say that lim a n = L; if for every ">0 there is an N>0 so that whenever n>n;ja n
More information8.5 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:
More information11.8 Power Series. Recall the geometric series. (1) x n = 1+x+x 2 + +x n +
11.8 1 11.8 Power Series Recall the geometric series (1) x n 1+x+x 2 + +x n + n As we saw in section 11.2, the series (1) diverges if the common ratio x > 1 and converges if x < 1. In fact, for all x (
More informationInfinite series, improper integrals, and Taylor series
Chapter Infinite series, improper integrals, and Taylor series. Determine which of the following sequences converge or diverge (a) {e n } (b) {2 n } (c) {ne 2n } (d) { 2 n } (e) {n } (f) {ln(n)} 2.2 Which
More informationThe 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.
More informationy = x 3 and y = 2x 2 x. 2x 2 x = x 3 x 3 2x 2 + x = 0 x(x 2 2x + 1) = 0 x(x 1) 2 = 0 x = 0 and x = (x 3 (2x 2 x)) dx
Millersville University Name Answer Key Mathematics Department MATH 2, Calculus II, Final Examination May 4, 2, 8:AM-:AM Please answer the following questions. Your answers will be evaluated on their correctness,
More informationJUST THE MATHS UNIT NUMBER DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) A.J.Hobson
JUST THE MATHS UNIT NUMBER.5 DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) by A.J.Hobson.5. Maclaurin s series.5. Standard series.5.3 Taylor s series.5.4 Exercises.5.5 Answers to exercises
More informationMath 143 Flash Cards. Divergence of a sequence {a n } {a n } diverges to. Sandwich Theorem for Sequences. Continuous Function Theorem for Sequences
Math Flash cards Math 143 Flash Cards Convergence of a sequence {a n } Divergence of a sequence {a n } {a n } diverges to Theorem (10.1) {a n } diverges to Sandwich Theorem for Sequences Theorem (10.1)
More information8.1 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
More informationc n (x a) n c 0 c 1 (x a) c 2 (x a) 2...
3 CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS 6. REVIEW OF POWER SERIES REVIEW MATERIAL Infinite series of constants, p-series, harmonic series, alternating harmonic series, geometric series, tests
More informationMath 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
More informationTaylor 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
More informationReview (11.1) 1. A sequence is an infinite list of numbers {a n } n=1 = a 1, a 2, a 3, The sequence is said to converge if lim
Announcements: Note that we have taking the sections of Chapter, out of order, doing section. first, and then the rest. Section. is motivation for the rest of the chapter. Do the homework questions from
More information1 Question related to polynomials
07-08 MATH00J Lecture 6: Taylor Series Charles Li Warning: Skip the material involving the estimation of error term Reference: APEX Calculus This lecture introduced Taylor Polynomial and Taylor Series
More informationMATH 137 : Calculus 1 for Honours Mathematics Instructor: Barbara Forrest Self Check #1: Sequences and Convergence
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:
More informationMATH 1231 MATHEMATICS 1B Calculus Section 4.4: Taylor & Power series.
MATH 1231 MATHEMATICS 1B 2010. For use in Dr Chris Tisdell s lectures. Calculus Section 4.4: Taylor & Power series. 1. What is a Taylor series? 2. Convergence of Taylor series 3. Common Maclaurin series
More informationRoot 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.
More informationSequences and Series
Sequences and Series Consider the following sum: 2 + 4 + 8 + 6 + + 2 i + The dots at the end indicate that the sum goes on forever. Does this make sense? Can we assign a numerical value to an infinite
More informationTaylor Series. Math114. March 1, Department of Mathematics, University of Kentucky. Math114 Lecture 18 1/ 13
Taylor Series Math114 Department of Mathematics, University of Kentucky March 1, 2017 Math114 Lecture 18 1/ 13 Given a function, can we find a power series representation? Math114 Lecture 18 2/ 13 Given
More informationRelationship Between Integration and Differentiation
Relationship Between Integration and Differentiation Fundamental Theorem of Calculus Philippe B. Laval KSU Today Philippe B. Laval (KSU) FTC Today 1 / 16 Introduction In the previous sections we defined
More informationMATH 1231 MATHEMATICS 1B CALCULUS. Section 5: - Power Series and Taylor Series.
MATH 1231 MATHEMATICS 1B CALCULUS. Section 5: - Power Series and Taylor Series. The objective of this section is to become familiar with the theory and application of power series and Taylor series. By
More informationMATH 6B Spring 2017 Vector Calculus II Study Guide Final Exam Chapters 8, 9, and Sections 11.1, 11.2, 11.7, 12.2, 12.3.
MATH 6B pring 2017 Vector Calculus II tudy Guide Final Exam Chapters 8, 9, and ections 11.1, 11.2, 11.7, 12.2, 12.3. Before starting with the summary of the main concepts covered in the quarter, here there
More informationTaylor 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
More informationSubsequences and Limsups. Some sequences of numbers converge to limits, and some do not. For instance,
Subsequences and Limsups Some sequences of numbers converge to limits, and some do not. For instance,,, 3, 4, 5,,... converges to 0 3, 3., 3.4, 3.4, 3.45, 3.459,... converges to π, 3,, 3.,, 3.4,... does
More informationClassnotes - MA Series and Matrices
Classnotes - MA-2 Series and Matrices Department of Mathematics Indian Institute of Technology Madras This classnote is only meant for academic use. It is not to be used for commercial purposes. For suggestions
More informationSection 5.8. Taylor Series
Difference Equations to Differential Equations Section 5.8 Taylor Series In this section we will put together much of the work of Sections 5.-5.7 in the context of a discussion of Taylor series. We begin
More informationAppendix A. Sequences and series. A.1 Sequences. Definition A.1 A sequence is a function N R.
Appendix A Sequences and series This course has for prerequisite a course (or two) of calculus. The purpose of this appendix is to review basic definitions and facts concerning sequences and series, which
More informationInfinite Series Summary
Infinite Series Summary () Special series to remember: Geometric series ar n Here a is the first term and r is the common ratio. When r
More informationTaylor and Maclaurin Series
Taylor and Maclaurin Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Background We have seen that some power series converge. When they do, we can think of them as
More informationInfinite Sequences and Series Section
A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics Infinite Sequences and Series Section 8.1-8.2 Dr. John Ehrke Department of Mathematics Fall 2012 Zeno s Paradox Achilles and
More informationMATH3283W LECTURE NOTES: WEEK 6 = 5 13, = 2 5, 1 13
MATH383W LECTURE NOTES: WEEK 6 //00 Recursive sequences (cont.) Examples: () a =, a n+ = 3 a n. The first few terms are,,, 5 = 5, 3 5 = 5 3, Since 5
More informationMath 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)
More informationName. 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
More informationTesting Series With Mixed Terms
Testing Series With Mixed Terms Philippe B. Laval Series with Mixed Terms 1. Introduction 2. Absolute v.s. Conditional Convergence 3. Alternating Series 4. The Ratio and Root Tests 5. Conclusion 1 Introduction
More informationSECTION 9.2: ARITHMETIC SEQUENCES and PARTIAL SUMS
(Chapter 9: Discrete Math) 9.11 SECTION 9.2: ARITHMETIC SEQUENCES and PARTIAL SUMS PART A: WHAT IS AN ARITHMETIC SEQUENCE? The following appears to be an example of an arithmetic (stress on the me ) sequence:
More informationPart 3.3 Differentiation Taylor Polynomials
Part 3.3 Differentiation 3..3.1 Taylor Polynomials Definition 3.3.1 Taylor 1715 and Maclaurin 1742) If a is a fixed number, and f is a function whose first n derivatives exist at a then the Taylor polynomial
More informationSequences and Summations
COMP 182 Algorithmic Thinking Sequences and Summations Luay Nakhleh Computer Science Rice University Chapter 2, Section 4 Reading Material Sequences A sequence is a function from a subset of the set of
More informationSequences and Series of Functions
Chapter 13 Sequences and Series of Functions These notes are based on the notes A Teacher s Guide to Calculus by Dr. Louis Talman. The treatment of power series that we find in most of today s elementary
More informationBecause 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
More informationBernoulli Numbers and their Applications
Bernoulli Numbers and their Applications James B Silva Abstract The Bernoulli numbers are a set of numbers that were discovered by Jacob Bernoulli (654-75). This set of numbers holds a deep relationship
More information3. Infinite Series. The Sum of a Series. A series is an infinite sum of numbers:
3. Infinite Series A series is an infinite sum of numbers: The individual numbers are called the terms of the series. In the above series, the first term is, the second term is, and so on. The th term
More informationModule 9 : Infinite Series, Tests of Convergence, Absolute and Conditional Convergence, Taylor and Maclaurin Series
Module 9 : Infinite Series, Tests of Convergence, Absolute and Conditional Convergence, Taylor and Maclaurin Series Lecture 26 : Absolute convergence [Section 261] Objectives In this section you will learn
More information1. (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
More informationTest 3 - Answer Key Version B
Student s Printed Name: Instructor: CUID: Section: Instructions: You are not permitted to use a calculator on any portion of this test. You are not allowed to use any textbook, notes, cell phone, laptop,
More informationMath 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
More informationFrom Calculus II: An infinite series is an expression of the form
MATH 3333 INTERMEDIATE ANALYSIS BLECHER NOTES 75 8. Infinite series of numbers From Calculus II: An infinite series is an expression of the form = a m + a m+ + a m+2 + ( ) Let us call this expression (*).
More informationGeometric Series and the Ratio and Root Test
Geometric Series and the Ratio and Root Test James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 5, 2018 Outline 1 Geometric Series
More informationConvergence of sequences and series
Convergence of sequences and series A sequence f is a map from N the positive integers to a set. We often write the map outputs as f n rather than f(n). Often we just list the outputs in order and leave
More informationSeries. Xinyu Liu. April 26, Purdue University
Series Xinyu Liu Purdue University April 26, 2018 Convergence of Series i=0 What is the first step to determine the convergence of a series? a n 2 of 21 Convergence of Series i=0 What is the first step
More informationTaylor Series. richard/math230 These notes are taken from Calculus Vol I, by Tom M. Apostol,
Taylor 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
More informationWeek 2: Sequences and Series
QF0: Quantitative Finance August 29, 207 Week 2: Sequences and Series Facilitator: Christopher Ting AY 207/208 Mathematicians have tried in vain to this day to discover some order in the sequence of prime
More informationEECS 1028 M: Discrete Mathematics for Engineers
EECS 1028 M: Discrete Mathematics for Engineers Suprakash Datta Office: LAS 3043 Course page: http://www.eecs.yorku.ca/course/1028 Also on Moodle S. Datta (York Univ.) EECS 1028 W 18 1 / 16 Sequences and
More informationSequences and infinite series
Sequences and infinite series D. DeTurck University of Pennsylvania March 29, 208 D. DeTurck Math 04 002 208A: Sequence and series / 54 Sequences The lists of numbers you generate using a numerical method
More informationSeries Solutions. 8.1 Taylor Polynomials
8 Series Solutions 8.1 Taylor Polynomials Polynomial functions, as we have seen, are well behaved. They are continuous everywhere, and have continuous derivatives of all orders everywhere. It also turns
More informationTAYLOR 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
More informationNotes on uniform convergence
Notes on uniform convergence Erik Wahlén erik.wahlen@math.lu.se January 17, 2012 1 Numerical sequences We begin by recalling some properties of numerical sequences. By a numerical sequence we simply mean
More informationMath 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
More informationSequences and Series. 256 Chapter 11 Sequences and Series. and then lim 1 1 = 1 0 = 1.
256 Chapter Sequences and Series Consider the following sum: Sequences and Series 2 + 4 + 8 + 6 + + 2 + i The dots at the end indicate that the sum goes on forever Does this make sense? Can we assign a
More informationThe 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?
More informationPolynomial Approximations and Power Series
Polynomial Approximations and Power Series June 24, 206 Tangent Lines One of the first uses of the derivatives is the determination of the tangent as a linear approximation of a differentiable function
More informationJuly 21 Math 2254 sec 001 Summer 2015
July 21 Math 2254 sec 001 Summer 2015 Section 8.8: Power Series Theorem: Let a n (x c) n have positive radius of convergence R, and let the function f be defined by this power series f (x) = a n (x c)
More informationThe 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
More informationLimits and Continuity
Chapter Limits and Continuity. Limits of Sequences.. The Concept of Limit and Its Properties A sequence { } is an ordered infinite list x,x,...,,... The n-th term of the sequence is, and n is the index
More informationSeries Solution of Linear Ordinary Differential Equations
Series Solution of Linear Ordinary Differential Equations Department of Mathematics IIT Guwahati Aim: To study methods for determining series expansions for solutions to linear ODE with variable coefficients.
More information