FINAL REVIEW FOR MATH The limit. a n. This definition is useful is when evaluating the limits; for instance, to show
|
|
- Willis Powell
- 6 years ago
- Views:
Transcription
1 FINAL REVIEW FOR MATH 500 SHUANGLIN SHAO. The it Define a n = A: For any ε > 0, there exists N N such that for any n N, a n A < ε. This definition is useful is when evaluating the its; for instance, to show 2n 3n + = 2 3. One can also apply the it theorems to evaluate the it as follows. 2n 3n + = n = 2 3. We talk about the it theorem for its. If a n = A and b n = B, then a n ± b n = A ± B a nb n = AB a n = A, if B 0. b n B The proof of this theorem is by use of the definition of its. There is also another useful theorem for evaluating the its, the squeezing theorem : If a n b n c n and a n = c n = A, then b n = A. This theorem can be also proven by using the definition of its. A typical example is as follows. sin n 2 = 0 n Date: April 24, 202.
2 because 0 sin n 2 n n, 0 = n = The monotone convergence theorem and the sequence The monotone convergence theorem says that, if a sequence {a n } is either monotone increasing and bounded above, or monotone decreasing and bounded below, then {a n } converges. We give an example to show how to apply this theorem. Question. Determine whether the sequence {a n }: a 0 = and a n+ = 3 + 2an converges or not. If it converges, then find the it. Proof. We first show that it is monotone increasing. I have seen two ways: the first way is by using the mathematics induction. Since a 0 =, then a = 3 + 2a 0 = 5. So a 0 a. We assume that a n a n. Then 3 + 2an 3 + 2a n. Then a n a n+. The second way I talked about it in class. We look at a n+ a n = 3 + 2a n 3 + 2a n = 2(a n a n ) 3 + 2an a n. The denominator is always positive. We see that the sign of a n+ a n is determined by the previous difference, a n a n. Inductively, we see that it is determined by a a 0, which is positive. So a n+ a n, for all n. We then show that it is bounded. We guess the bound is 3. We also prove it by mathematics induction. a 0 = 3. We assume that a n 3. Then a n+ = 3 + 2a n = 3. 2
3 So a n 3 for all n. Then we apply the it theorems to get the it. A = 3 + 2A, i.e., A 2 2A 3 = 0. Then But A > 0, so A = 3 or A =. A = Series The convergence of a series k= a k is determined by its partial sum, n S n = a k = a + a 2 + a n. k= If {S n } converges, then a k converges and S n = a k. Otherwise, we say that the series diverges. The definition often provides a way to tell whether a series converges or diverges. There are several well-known examples of convergent or divergent examples: the following series, n=, p >, and np n=2 n(ln n) p, p >. They are convergent by comparing to the integrals, both of which are finite because p >. n= x dx and p On the other hand, the following series are divergent,, 0 < p, and np n(ln n) p, 0 < p. n=2 They are divergent by comparing to the integrals, both of which are infinity because 0 < p. The geometric series is convergent. r n for r <. n= 3 x dx and p x(ln x) p dx, x(ln x) p dx,
4 We can prove this by setting Then S n = n r k. k= ( r)s n = (r + r 2 + r n ) (r r n + r n+ ) = r r n+. Since r <, S n = r rn+ r S n = r r n+ r = r r. We know that the harmonic series n= n is divergent. However if we add signs to each term ( )n n to turn it into n, then the series ( ) n n= n is convergent. In this regard, there is a general theorem: Suppose the sequence {a n } satisfying a a 2 a n 0 and n 0 a n = 0, then the series n= ( )n a n is convergent. A consequence that n= a n is convergent is that the tail terms go to zero as n goes to infinity, i.e., () a n = 0. This is a necessary not sufficient condition for a convergent series, for example, for n= n, n = 0 but the series n= n is divergent. However the equation () can be used as a criterion to tell whether a series converges or not: for n= a n, if a n 0, then n= a n is divergent. An example is, n= 2n is divergent because 2 n =. Another example n= ( + n ) is divergent because + n = series of functions In the previous section, we talk about several theorems about the series n= a n. By adding a variable x, the series of functions n= a nx n is also a kind of series with terms a n x n. Obviously it is convergent if x = 0. In general, there is a quantity associated to each series of functions, the radius of convergence ρ, such that, for x < ρ, the series is absolutely convergent; for x > ρ, it is divergent. However, for x = ρ, we need to study it case by case. We use the ratio test to find ρ. For example, for which values of x, does the series x n n converge? Proof. For this example, a n = n. 4
5 Then a n+ a n n = =, i.e., q =. (n + ) Hence the radius of convergence ρ = q =. That is to say, for x <, the series n= a nx n is absolutely convergent. For x >, the series n= a nx n is divergent. For x =, x = or x =. For x =, the series becomes n, which diverges. However for x =, the series becomes ( ) n n= n, which is convergent because it is a series of alternating terms. To conclude, the series is convergent on [, ), and divergent for x > and x =. 5. Limits of functions, continuity and uniform continuity Let f : D R, and a R. We say that x a f(x) = A, if for every sequence {x n } D satisfying x n a, f(x n) = A. Note that the point a may not be in the domain D. If there exists two sequences {x n } and {y n } both converging to a, and its n a f(x n ) and f(y n ) are not equal, then x a f(x) does not exist. This is a typical way to show that a it does not exist. An example, f(x) = x : R \ {0} R, in this case, f(x) does not exist. x x 0 The continuity of functions can be defined in a similar way. We say that f is continuous at x = a, if for every sequence {x n } D satisfying x n a, f(x n) = f(a). Note that a is in the domain D. If f is continuous at every point of D, then f is continuous on D. We have learned two theorems about continuous functions. One is the intermediate value theorem, and the other is the maximum principle. In the final, you are not required to know the proofs of these two theorems, but you need to know how to apply them to solve the problems. There is another way to phrase the continuity of functions, which we call the ɛ δ formulation of continuity of functions. We say that f is continuous 5
6 at x = a: for any ɛ > 0, there exists δ > 0 such that for x a < δ, Here δ may depend on ɛ and a. f(x) f(a) < ɛ. The second definition has the advantage of generalizing to uniform continuity of functions. We say that f is uniformly continuous on D: for any ɛ > 0, there exists δ > 0 such that for x y < δ, f(x) f(y) < ɛ. In general, the uniform continuity of functions implies the continuity of functions, but the converse is not true. For instance, f(x) = x 2 : R R is continuous on R but not uniformly continuous. However on a closed interval, say [a, b], they are equivalent. 6. differentiability Having seen continuity of functions, we talk about another property of functions, differentiability. We say that f is differentiable at a point a, if the following it exists, If the it exists, we denote h 0 f(a + h) f(a). h f f(a + h) f(a) (a) =. h 0 h The definition is important when we want to say whether a function is differentiable at a point; also it will gives the formula of the derivatives. For instance, f(x) = x n with n, is differentiable at every point x and f (x) = nx n. Another example is, for f(x) = sin x, f (x) = cos x. The most common example of non-differentiable functions is f(x) = x ; for this function, although it is continuous at x = 0, we know that it is not differentiable at x = 0. Related to differentiability, there are two important theorems, Rolle s theorem and the mean value theorem. The proof of Rolle s theorem makes use of the maximum principle we just mentioned. Roughly speaking, for continuous functions over [a, b], there exists c and d in [a, b] such that f(c) = max a x b f(x), f(d) = min f(x). a x b At these points, one can show that f = 0. The Rolle Theorem can be used to prove the mean value theorem. 6
7 7. integrability Let f be a bounded real function. We say that f is Riemann integrable on [a, b], if L(f) = U(f), where, L(f) = sup{l P (f)}, U(f) = inf {U P (f)}. P P Here P is a partition of [a, b]. We have shown that continuous functions or monotone functions over [a, b] are integrable. The Dirichlet function f is not Riemann integrable, where {, for irrational x; f(x) = 0, for rational x. Related to Riemann integrals, the fundamental theorem of calculus is important. Let f be a continuous function on [a, b]. Then is differentiable at x (a, b) and F (x) = F (x) = f(x), and b a x a f(t)dt f(t)dt = F (b) F (a). Note that F is called the anti-derivative of f. The last equation in the Fundament Theorem of Calculus tells a way to compute integrals b a f(t)dt: we just need to find what are the antiderivative of f. For example, π 0 sin xdx = ( cos π) + cos 0 = 2. This is because the antiderivative of sin x is cos x. Department of Mathematics, KU, Lawrence, KS address: slshao@math.ku.edu 7
MIDTERM REVIEW FOR MATH The limit
MIDTERM REVIEW FOR MATH 500 SHUANGLIN SHAO. The limit Define lim n a n = A: For any ε > 0, there exists N N such that for any n N, a n A < ε. The key in this definition is to realize that the choice of
More informationSolutions Final Exam May. 14, 2014
Solutions Final Exam May. 14, 2014 1. (a) (10 points) State the formal definition of a Cauchy sequence of real numbers. A sequence, {a n } n N, of real numbers, is Cauchy if and only if for every ɛ > 0,
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 informationSolutions Final Exam May. 14, 2014
Solutions Final Exam May. 14, 2014 1. Determine whether the following statements are true or false. Justify your answer (i.e., prove the claim, derive a contradiction or give a counter-example). (a) (10
More informationLogical Connectives and Quantifiers
Chapter 1 Logical Connectives and Quantifiers 1.1 Logical Connectives 1.2 Quantifiers 1.3 Techniques of Proof: I 1.4 Techniques of Proof: II Theorem 1. Let f be a continuous function. If 1 f(x)dx 0, then
More informationMATH 409 Advanced Calculus I Lecture 10: Continuity. Properties of continuous functions.
MATH 409 Advanced Calculus I Lecture 10: Continuity. Properties of continuous functions. Continuity Definition. Given a set E R, a function f : E R, and a point c E, the function f is continuous at c if
More information1 Definition of the Riemann integral
MAT337H1, Introduction to Real Analysis: notes on Riemann integration 1 Definition of the Riemann integral Definition 1.1. Let [a, b] R be a closed interval. A partition P of [a, b] is a finite set of
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 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 informationMATH 131A: REAL ANALYSIS (BIG IDEAS)
MATH 131A: REAL ANALYSIS (BIG IDEAS) Theorem 1 (The Triangle Inequality). For all x, y R we have x + y x + y. Proposition 2 (The Archimedean property). For each x R there exists an n N such that n > x.
More informationconverges as well if x < 1. 1 x n x n 1 1 = 2 a nx n
Solve the following 6 problems. 1. Prove that if series n=1 a nx n converges for all x such that x < 1, then the series n=1 a n xn 1 x converges as well if x < 1. n For x < 1, x n 0 as n, so there exists
More informationEntrance Exam, Real Analysis September 1, 2017 Solve exactly 6 out of the 8 problems
September, 27 Solve exactly 6 out of the 8 problems. Prove by denition (in ɛ δ language) that f(x) = + x 2 is uniformly continuous in (, ). Is f(x) uniformly continuous in (, )? Prove your conclusion.
More informationREAL VARIABLES: PROBLEM SET 1. = x limsup E k
REAL VARIABLES: PROBLEM SET 1 BEN ELDER 1. Problem 1.1a First let s prove that limsup E k consists of those points which belong to infinitely many E k. From equation 1.1: limsup E k = E k For limsup E
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 informationDefinitions & Theorems
Definitions & Theorems Math 147, Fall 2009 December 19, 2010 Contents 1 Logic 2 1.1 Sets.................................................. 2 1.2 The Peano axioms..........................................
More informationSection 11.1 Sequences
Math 152 c Lynch 1 of 8 Section 11.1 Sequences A sequence is a list of numbers written in a definite order: a 1, a 2, a 3,..., a n,... Notation. The sequence {a 1, a 2, a 3,...} can also be written {a
More informationA Short Journey Through the Riemann Integral
A Short Journey Through the Riemann Integral Jesse Keyton April 23, 2014 Abstract An introductory-level theory of integration was studied, focusing primarily on the well-known Riemann integral and ending
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 informationSOME QUESTIONS FOR MATH 766, SPRING Question 1. Let C([0, 1]) be the set of all continuous functions on [0, 1] endowed with the norm
SOME QUESTIONS FOR MATH 766, SPRING 2016 SHUANGLIN SHAO Question 1. Let C([0, 1]) be the set of all continuous functions on [0, 1] endowed with the norm f C = sup f(x). 0 x 1 Prove that C([0, 1]) is a
More informationMATH 409 Advanced Calculus I Lecture 9: Limit supremum and infimum. Limits of functions.
MATH 409 Advanced Calculus I Lecture 9: Limit supremum and infimum. Limits of functions. Limit points Definition. A limit point of a sequence {x n } is the limit of any convergent subsequence of {x n }.
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 informationMath 117: Honours Calculus I Fall, 2002 List of Theorems. a n k b k. k. Theorem 2.1 (Convergent Bounded) A convergent sequence is bounded.
Math 117: Honours Calculus I Fall, 2002 List of Theorems Theorem 1.1 (Binomial Theorem) For all n N, (a + b) n = n k=0 ( ) n a n k b k. k Theorem 2.1 (Convergent Bounded) A convergent sequence is bounded.
More informationMathematical Methods for Physics and Engineering
Mathematical Methods for Physics and Engineering Lecture notes for PDEs Sergei V. Shabanov Department of Mathematics, University of Florida, Gainesville, FL 32611 USA CHAPTER 1 The integration theory
More informationChapter 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
More informationPrinciple of Mathematical Induction
Advanced Calculus I. Math 451, Fall 2016, Prof. Vershynin Principle of Mathematical Induction 1. Prove that 1 + 2 + + n = 1 n(n + 1) for all n N. 2 2. Prove that 1 2 + 2 2 + + n 2 = 1 n(n + 1)(2n + 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 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 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 informationM2PM1 Analysis II (2008) Dr M Ruzhansky List of definitions, statements and examples Preliminary version
M2PM1 Analysis II (2008) Dr M Ruzhansky List of definitions, statements and examples Preliminary version Chapter 0: Some revision of M1P1: Limits and continuity This chapter is mostly the revision of Chapter
More informationArkansas Tech University MATH 2924: Calculus II Dr. Marcel B. Finan
Arkansas Tech University MATH 2924: Calculus II Dr. Marcel B. Finan 8. Sequences We start this section by introducing the concept of a sequence and study its convergence. Convergence of Sequences. An infinite
More informationM2P1 Analysis II (2005) Dr M Ruzhansky List of definitions, statements and examples. Chapter 1: Limits and continuity.
M2P1 Analysis II (2005) Dr M Ruzhansky List of definitions, statements and examples. Chapter 1: Limits and continuity. This chapter is mostly the revision of Chapter 6 of M1P1. First we consider functions
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 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 informationPower 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
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 informationTopics Covered in Calculus BC
Topics Covered in Calculus BC Calculus BC Correlation 5 A Functions, Graphs, and Limits 1. Analysis of graphs 2. Limits or functions (including one sides limits) a. An intuitive understanding of the limiting
More informationWe have been going places in the car of calculus for years, but this analysis course is about how the car actually works.
Analysis I We have been going places in the car of calculus for years, but this analysis course is about how the car actually works. Copier s Message These notes may contain errors. In fact, they almost
More informationMath 106 Fall 2014 Exam 2.1 October 31, ln(x) x 3 dx = 1. 2 x 2 ln(x) + = 1 2 x 2 ln(x) + 1. = 1 2 x 2 ln(x) 1 4 x 2 + C
Math 6 Fall 4 Exam. October 3, 4. The following questions have to do with the integral (a) Evaluate dx. Use integration by parts (x 3 dx = ) ( dx = ) x3 x dx = x x () dx = x + x x dx = x + x 3 dx dx =
More informationMATH 104 : Final Exam
MATH 104 : Final Exam 10 May, 2017 Name: You have 3 hours to answer the questions. You are allowed one page (front and back) worth of notes. The page should not be larger than a standard US letter size.
More informationIntroduction to Proofs in Analysis. updated December 5, By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION
Introduction to Proofs in Analysis updated December 5, 2016 By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION Purpose. These notes intend to introduce four main notions from
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 informationd(x n, x) d(x n, x nk ) + d(x nk, x) where we chose any fixed k > N
Problem 1. Let f : A R R have the property that for every x A, there exists ɛ > 0 such that f(t) > ɛ if t (x ɛ, x + ɛ) A. If the set A is compact, prove there exists c > 0 such that f(x) > c for all x
More informationLecture 10. Riemann-Stieltjes Integration
Lecture 10 Riemann-Stieltjes Integration In this section we will develop the basic definitions and some of the properties of the Riemann-Stieltjes Integral. The development will follow that of the Darboux
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 informationCH 2: Limits and Derivatives
2 The tangent and velocity problems CH 2: Limits and Derivatives the tangent line to a curve at a point P, is the line that has the same slope as the curve at that point P, ie the slope of the tangent
More informationSection 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
More informationBounded Derivatives Which Are Not Riemann Integrable. Elliot M. Granath. A thesis submitted in partial fulfillment of the requirements
Bounded Derivatives Which Are Not Riemann Integrable by Elliot M. Granath A thesis submitted in partial fulfillment of the requirements for graduation with Honors in Mathematics. Whitman College 2017 Certificate
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 informationAdvanced Calculus II Unit 7.3: 7.3.1a, 7.3.3a, 7.3.6b, 7.3.6f, 7.3.6h Unit 7.4: 7.4.1b, 7.4.1c, 7.4.2b, 7.4.3, 7.4.6, 7.4.7
Advanced Calculus II Unit 73: 73a, 733a, 736b, 736f, 736h Unit 74: 74b, 74c, 74b, 743, 746, 747 Megan Bryant October 9, 03 73a Prove the following: If lim p a = A, for some p >, then a converges absolutely
More informationAnalysis/Calculus Review Day 2
Analysis/Calculus Review Day 2 AJ Friend ajfriend@stanford.edu Arvind Saibaba arvindks@stanford.edu Institute of Computational and Mathematical Engineering Stanford University September 20, 2011 Continuity
More informationMATH 2400: PRACTICE PROBLEMS FOR EXAM 1
MATH 2400: PRACTICE PROBLEMS FOR EXAM 1 PETE L. CLARK 1) Find all real numbers x such that x 3 = x. Prove your answer! Solution: If x 3 = x, then 0 = x 3 x = x(x + 1)(x 1). Earlier we showed using the
More informationMathematics 324 Riemann Zeta Function August 5, 2005
Mathematics 324 Riemann Zeta Function August 5, 25 In this note we give an introduction to the Riemann zeta function, which connects the ideas of real analysis with the arithmetic of the integers. Define
More informationREAL ANALYSIS II: PROBLEM SET 2
REAL ANALYSIS II: PROBLEM SET 2 21st Feb, 2016 Exercise 1. State and prove the Inverse Function Theorem. Theorem Inverse Function Theorem). Let f be a continuous one to one function defined on an interval,
More informationAnalysis Finite and Infinite Sets The Real Numbers The Cantor Set
Analysis Finite and Infinite Sets Definition. An initial segment is {n N n n 0 }. Definition. A finite set can be put into one-to-one correspondence with an initial segment. The empty set is also considered
More informationMATH 18.01, FALL PROBLEM SET #5 SOLUTIONS (PART II)
MATH 8, FALL 7 - PROBLEM SET #5 SOLUTIONS (PART II (Oct ; Antiderivatives; + + 3 7 points Recall that in pset 3A, you showed that (d/dx tanh x x Here, tanh (x denotes the inverse to the hyperbolic tangent
More informationAP Calculus BC Scope & Sequence
AP Calculus BC Scope & Sequence Grading Period Unit Title Learning Targets Throughout the School Year First Grading Period *Apply mathematics to problems in everyday life *Use a problem-solving model that
More informationMath 320: Real Analysis MWF 1pm, Campion Hall 302 Homework 4 Solutions Please write neatly, and in complete sentences when possible.
Math 320: Real Analysis MWF pm, Campion Hall 302 Homework 4 Solutions Please write neatly, and in complete sentences when possible. Do the following problems from the book: 2.6.3, 2.7.4, 2.7.5, 2.7.2,
More informationLimits and continuity
CHAPTER 4 Limits and continuity Our first goal is to define and understand lim f(x) =L. Here f : D R where D R. We want the definition to mean roughly, as x gets close to a then f(x) iscloseto L. Perhaps
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 informationCopyright c 2007 Jason Underdown Some rights reserved. statement. sentential connectives. negation. conjunction. disjunction
Copyright & License Copyright c 2007 Jason Underdown Some rights reserved. statement sentential connectives negation conjunction disjunction implication or conditional antecedant & consequent hypothesis
More informationRepresentation 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
More informationSection 4.1 Relative Extrema 3 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 4.1 Relative Extrema 3 Lectures College of Science MATHS 101: Calculus I (University of Bahrain) Extrema 1 / 16 Application of Differentiation One of the most important applications of differential
More informationQuick Tour of the Topology of R. Steven Hurder, Dave Marker, & John Wood 1
Quick Tour of the Topology of R Steven Hurder, Dave Marker, & John Wood 1 1 Department of Mathematics, University of Illinois at Chicago April 17, 2003 Preface i Chapter 1. The Topology of R 1 1. Open
More informationMATH 409 Advanced Calculus I Lecture 11: More on continuous functions.
MATH 409 Advanced Calculus I Lecture 11: More on continuous functions. Continuity Definition. Given a set E R, a function f : E R, and a point c E, the function f is continuous at c if for any ε > 0 there
More informationvan Rooij, Schikhof: A Second Course on Real Functions
vanrooijschikhof.tex April 25, 2018 van Rooij, Schikhof: A Second Course on Real Functions Notes from [vrs]. Introduction A monotone function is Riemann integrable. A continuous function is Riemann integrable.
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 informationn=1 ( 2 3 )n (a n ) converges by direct comparison to
. (a) n = a n converges, so we know that a n =. Therefore, for n large enough we know that a n
More informationMATH 162. Midterm 2 ANSWERS November 18, 2005
MATH 62 Midterm 2 ANSWERS November 8, 2005. (0 points) Does the following integral converge or diverge? To get full credit, you must justify your answer. 3x 2 x 3 + 4x 2 + 2x + 4 dx You may not be able
More information1. Let A R be a nonempty set that is bounded from above, and let a be the least upper bound of A. Show that there exists a sequence {a n } n N
Applied Analysis prelim July 15, 216, with solutions Solve 4 of the problems 1-5 and 2 of the problems 6-8. We will only grade the first 4 problems attempted from1-5 and the first 2 attempted from problems
More information(b) Prove that the following function does not tend to a limit as x tends. is continuous at 1. [6] you use. (i) f(x) = x 4 4x+7, I = [1,2]
TMA M208 06 Cut-off date 28 April 2014 (Analysis Block B) Question 1 (Unit AB1) 25 marks This question tests your understanding of limits, the ε δ definition of continuity and uniform continuity, and your
More informationInternational Competition in Mathematics for Universtiy Students in Plovdiv, Bulgaria 1994
International Competition in Mathematics for Universtiy Students in Plovdiv, Bulgaria 1994 1 PROBLEMS AND SOLUTIONS First day July 29, 1994 Problem 1. 13 points a Let A be a n n, n 2, symmetric, invertible
More informationAn idea how to solve some of the problems. diverges the same must hold for the original series. T 1 p T 1 p + 1 p 1 = 1. dt = lim
An idea how to solve some of the problems 5.2-2. (a) Does not converge: By multiplying across we get Hence 2k 2k 2 /2 k 2k2 k 2 /2 k 2 /2 2k 2k 2 /2 k. As the series diverges the same must hold for the
More information2.8 Linear Approximations and Differentials
Arkansas Tech University MATH 294: Calculus I Dr. Marcel B. Finan 2.8 Linear Approximations and Differentials In this section we approximate graphs by tangent lines which we refer to as tangent line approximations.
More informationMath 409 Final Exam Solutions May 9, 2005
Math 49 Final Exam Solutions May 9, 25 1. 25) State and prove the monotone convergence theorem. Then use it to show that the recursive sequence defined below converges to some number l. What must l equal?
More informationAP Calculus BC Syllabus Course Overview
AP Calculus BC Syllabus Course Overview Textbook Anton, Bivens, and Davis. Calculus: Early Transcendentals, Combined version with Wiley PLUS. 9 th edition. Hoboken, NJ: John Wiley & Sons, Inc. 2009. Course
More informationFoundations of Calculus. November 18, 2014
Foundations of Calculus November 18, 2014 Contents 1 Conic Sections 3 11 A review of the coordinate system 3 12 Conic Sections 4 121 Circle 4 122 Parabola 5 123 Ellipse 5 124 Hyperbola 6 2 Review of Functions
More informationMATH 1207 R02 FINAL SOLUTION
MATH 7 R FINAL SOLUTION SPRING 6 - MOON Write your answer neatly and show steps. Except calculators, any electronic devices including laptops and cell phones are not allowed. () Let f(x) = x cos x. (a)
More informationLinearization and Extreme Values of Functions
Linearization and Extreme Values of Functions 3.10 Linearization and Differentials Linear or Tangent Line Approximations of function values Equation of tangent to y = f(x) at (a, f(a)): Tangent line approximation
More informationMa 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
More informationWorksheet 7, Math 10560
Worksheet 7, Math 0560 You must show all of your work to receive credit!. Determine whether the following series and sequences converge or diverge, and evaluate if they converge. If they diverge, you must
More informationLemma 15.1 (Sign preservation Lemma). Suppose that f : E R is continuous at some a R.
15. Intermediate Value Theorem and Classification of discontinuities 15.1. Intermediate Value Theorem. Let us begin by recalling the definition of a function continuous at a point of its domain. Definition.
More informationLIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS
LIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS RECALL: VERTICAL ASYMPTOTES Remember that for a rational function, vertical asymptotes occur at values of x = a which have infinite its (either positive or
More informationSolutions: Problem Set 4 Math 201B, Winter 2007
Solutions: Problem Set 4 Math 2B, Winter 27 Problem. (a Define f : by { x /2 if < x
More informationMATH NEW HOMEWORK AND SOLUTIONS TO PREVIOUS HOMEWORKS AND EXAMS
MATH. 4433. NEW HOMEWORK AND SOLUTIONS TO PREVIOUS HOMEWORKS AND EXAMS TOMASZ PRZEBINDA. Final project, due 0:00 am, /0/208 via e-mail.. State the Fundamental Theorem of Algebra. Recall that a subset K
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 informationMATH 5640: Fourier Series
MATH 564: Fourier Series Hung Phan, UMass Lowell September, 8 Power Series A power series in the variable x is a series of the form a + a x + a x + = where the coefficients a, a,... are real or complex
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 informationREVIEW OF ESSENTIAL MATH 346 TOPICS
REVIEW OF ESSENTIAL MATH 346 TOPICS 1. AXIOMATIC STRUCTURE OF R Doğan Çömez The real number system is a complete ordered field, i.e., it is a set R which is endowed with addition and multiplication operations
More informationInduction, sequences, limits and continuity
Induction, sequences, limits and continuity Material covered: eclass notes on induction, Chapter 11, Section 1 and Chapter 2, Sections 2.2-2.5 Induction Principle of mathematical induction: Let P(n) be
More informationMath 231E, Lecture 25. Integral Test and Estimating Sums
Math 23E, Lecture 25. Integral Test and Estimating Sums Integral Test The definition of determining whether the sum n= a n converges is:. Compute the partial sums s n = a k, k= 2. Check that s n is a convergent
More informationMath 1552: Integral Calculus Final Exam Study Guide, Spring 2018
Math 55: Integral Calculus Final Exam Study Guide, Spring 08 PART : Concept Review (Note: concepts may be tested on the exam in the form of true/false or short-answer questions.). Complete each statement
More information(e) 2 (f) 2. (c) + (d). Limits at Infinity. 2.5) 9-14,25-34,41-43,46-47,56-57, (c) (d) 2
Math 150A. Final Review Answers, Spring 2018. Limits. 2.2) 7-10, 21-24, 28-1, 6-8, 4-44. 1. Find the values, or state they do not exist. (a) (b) 1 (c) DNE (d) 1 (e) 2 (f) 2 (g) 2 (h) 4 2. lim f(x) = 2,
More informationNumerical Hestock Integration-Lecture Note 1
Numerical Hestock Integration-Lecture Note Wei-Chi YANG wyang@radford.edu Department of Mathematics and Statistics Radford University Radford, VA USA Abstract This is a brief version of the ejmt paper,
More informationWed. Sept 28th: 1.3 New Functions from Old Functions: o vertical and horizontal shifts o vertical and horizontal stretching and reflecting o
Homework: Appendix A: 1, 2, 3, 5, 6, 7, 8, 11, 13-33(odd), 34, 37, 38, 44, 45, 49, 51, 56. Appendix B: 3, 6, 7, 9, 11, 14, 16-21, 24, 29, 33, 36, 37, 42. Appendix D: 1, 2, 4, 9, 11-20, 23, 26, 28, 29,
More informationAnalysis Qualifying Exam
Analysis Qualifying Exam Spring 2017 Problem 1: Let f be differentiable on R. Suppose that there exists M > 0 such that f(k) M for each integer k, and f (x) M for all x R. Show that f is bounded, i.e.,
More informationAbsolute and Local Extrema
Extrema of Functions We can use the tools of calculus to help us understand and describe the shapes of curves. Here is some of the data that derivatives f (x) and f (x) can provide about the shape of the
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 informationChapter 8: Taylor s theorem and L Hospital s rule
Chapter 8: Taylor s theorem and L Hospital s rule Theorem: [Inverse Mapping Theorem] Suppose that a < b and f : [a, b] R. Given that f (x) > 0 for all x (a, b) then f 1 is differentiable on (f(a), f(b))
More information10/9/10. The line x = a is a vertical asymptote of the graph of a function y = f(x) if either. Definitions and Theorems.
Definitions and Theorems Introduction Unit 2 Limits and Continuity Definition - Vertical Asymptote Definition - Horizontal Asymptote Definition Continuity Unit 3 Derivatives Definition - Derivative Definition
More information5.3 Definite Integrals and Antiderivatives
5.3 Definite Integrals and Antiderivatives Objective SWBAT use properties of definite integrals, average value of a function, mean value theorem for definite integrals, and connect differential and integral
More information