Hanoi University of Technology Faculty of Applied mathematics and informatics Advanced Training Program Lecture on INFINITE SERIES AND DIFFERENTIAL EQUATIONS Dr. Nguyen Thieu Huy Ha Noi-2009 Nguyen Thieu Huy Preface The Lecture on infinite series and differential equations is written for students of Advanced Training Programs of Mechatronics (from California State University–CSU Chico) and Material Science (from University of Illinois- UIUC). To prepare for the manuscript of this lecture, we have to combine not only the two syllabuses of two courses on Differential Equations (Math 260 of CSU Chico and Math 385 of UIUC), but also the part of infinite series that should have been given in Calculus I and II according to the syllabuses of the CSU and UIUC (the Faculty of Applied Mathematics and Informatics of HUT decided to integrate the knowledge of infinite series with the differential equations in the same syllabus). Therefore, this lecture provides the most important modules of knowledge which are given in all syllabuses.
This lecture is intended for engineering students and others who require a working knowledge of differential equations and series; included are technique and applications of differential equations and infinite series. Since many physical laws and relations appear mathematically in the form of differential equations, such equations are of fundamental importance in engineering mathematics. Therefore, the main objective of this course is to help students to be familiar with various physical and geometrical problems that lead to differential equations and to provide students with the most important standard methods for solving such equations. I would like to thank Dr.
Tran Xuan Tiep for his reading and reviewing of the manuscript. I would like to express my love and gratefulness to my wife Dr. Vu Thi Ngoc Ha for her constant support and inspiration during the preparation of the lecture. Hanoi, April 4, 2009 Dr.
Nguyen Thieu Huy 1 Lecture on Infinite Series and Differential Equations Content CHAPTER 1: INFINITE SERIES. Definitions of Infinite Series and Fundamental Facts. Tests for Convergence and Divergence of Series of Constants. Theorem on Absolutely Convergent Series.
9 CHAPTER 2: INFINITE SEQUENCES AND SERIES OF FUNCTIONS. Basic Concepts of Sequences and Series of Functions. Theorems on uniformly convergent series. 22 CHAPTER 3: BASIC CONCEPT OF DIFFERENTIAL EQUATIONS.
Examples of Differential Equations. Definitions and Related Concepts. 30 CHAPTER 4: SOLUTIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS. Modelling: Electric Circuits.
Existence and Uniqueness Theorem. 40 CHAPTER 5: SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS. Definitions and Notations. Theory for Solutions of Linear Homogeneous Equations.
Homogeneous Equations with Constant Coefficients. Modelling: Free Oscillation (Mass-spring problem). Nonhomogeneous Equations: Method of Undetermined Coefficients. Variation of Parameters.
Modelling: Forced Oscillation. Power Series Solutions. 66 CHAPTER 6: Laplace Transform. Definition and Domain.
Applications to Differential Equations. 75 Tables of Laplace Transform. 80 2 Nguyen Thieu Huy CHAPTER 1: INFINITE SERIES The early developers of the calculus, including Newton and Leibniz, were well aware of the importance of infinite series. The values of many functions such as sine and cosine were geometrically obtainable only in special cases.
Infinite series provided a way of developing extensive tables of values for them. This chapter begins with a statement of what is meant by infinite series, then the question of when these sums can be assigned values is addressed. Much information can be obtained by exploring infinite sums of constant terms; however, the eventual objective in analysis is to introduce series that depend on variables. This presents the possibility of representing functions by series.
Afterward, the question of how continuity, differentiability, and integrability play a role can be examined. The question of dividing a line segment into infinitesimal parts has stimulated the imaginations of philosophers for a very long time. In a corruption of a paradox introduce by Zeno of Elea (in the fifth century B.) a dimensionless frog sits on the end of a one- dimensional log of unit length. The frog jumps halfway, and then halfway and halfway ad infinitum.
The question is whether the frog ever reaches the other end. Mathematically, an unending sum, is suggested. "Common sense" tells us that the sum must approach one even though that value is never attained. We can form sequences of partial sums and then examine the limit.
This returns us to Calculus I and the modern manner of thinking about the infinitesimal. In this chapter, consideration of such sums launches us on the road to the theory of infinite series. Definitions of Infinite Series and Fundamental Facts 1. Let {un} be a sequence of real numbers.
Then, the formal sum (1) is an infinite series. n Its value, if one exists, is the limit of the sequence of partial sums {Sn= u k }n1 k 1 3 Lecture on Infinite Series and Differential Equations If the limit exists, the series is said to converge to that sum, S. If the limit does not exist, the series is said to diverge. Sometimes the character of a series is obvious.
For example, the series generated by the frog on the log surely converges, while n diverges. On the other hand, n 1 the variable series raises questions. This series may be obtained by carrying out the division 1/(1-x). If -1 < x < 1, the sums Sn yields an approximations to 1/(1-x), passing to the limit, it is the exact value.
The indecision arises for x = -1. Some very great mathematicians, including Leonard Euler, thought that S should be equal to 1/2, as is obtained by substituting -1 into 1/(1-x). The problem with this conclusion arises with examination of 1 -1 + 1 -1+ 1 -1 + • • • and observation that appropriate associations can produce values of 1 or 0. Imposition of the condition of uniqueness for convergence put this series in the category of divergent and eliminated such possibility of ambiguity in other cases.
If u n converges, then limu n =0. The converse, however, is not necessarily true, i., if n n 1 limu n =0, u n may or may not converge. It follows that if the nth term of a series does not n n 1 approach zero, the series is divergent. Multiplication of each term of a series by a constant different from zero does not affect the convergence or divergence.
Removal (or addition) of a finite number of terms from (or to) a series does not affect the convergence or divergence. Tests for Convergence and Divergence of Series of Constants More often than not, exact values of infinite series cannot be obtained. Thus, the search turns toward information about the series. In particular, its convergence or divergence comes in question.
The following tests aid in discovering this information. 4 Nguyen Thieu Huy 2.1 Comparison test for series of non-negative terms. PROOF of Comparison test: (a) Let 0≤um≤ v n, n = 1, 2, 3,. Then, let Sn = u1 + u2+…+ un; n 1 n Tn=v1+v2+…+vn.
Since v n converges, limn->∞Tn exists and equals T, say. Also, since vn ≥ 0, Tn ≤T. n 1 Then Sn =u1+ u2 + •••+un ≤ v1+ v2 + ••• + vn =T or 0 ≤ Sn ≤ T. Thus {Sn} is a bounded monotonic increasing sequence and must have a limit, i.
(b) The proof of (b) is left for the reader as an exercise.2 The Limit-Comparison or Quotient Test for series of non-negative terms. 5 Lecture on Infinite Series and Differential Equations 1 sin 1 1 n 1 EXAMPLE: sin n converges, since sin n >0, lim 2 n 1 2 =1 and n converges. n 1 2 n1 2 n 2 This test is related to the comparison test and is often a very useful alternative to it. In particular, taking vn = l/np, we have the following theorem 2.3 Integral test for series of non-negative terms.
PROOF of Integral test: 6 Nguyen Thieu Huy 2.4 Alternating series test: An alternating series is one whose successive terms are alternately positive and negative. An alternating series un converges if the following two conditions are satisfied. PROOF: Let an be an alternating series satisfying the above conditions (a) and (b).5 Absolute and conditional convergence. Definition: The series u n is called absolutely convergent if | u n | converges.
If un n 1 n 1 n 1 converges but | u n | diverges, then un is called conditionally convergent. n 1 n 1 Lemma: The absolutely convergent series is convergent. PROOF: 7 Lecture on Infinite Series and Differential Equations 2.6 Ratio (D’Alembert) Test: Proof: a) Since L<1, we can take an ε > 0 such that 0<L+ ε<1. Then there exists an n0 such u that n1 <L+ ε for all n≥N.
Therefore, it follows that |un+1|<|un|(L+ ε) for all n≥N. Hence, un |un|<|un-1|(L+ ε)< |un-2|(L+ ε)2<…<|uN|(L+ ε)n-N for all n>N. Since | u N | ( L ) is convergent, it follows that | u n | is convergent by comparison n n 1 n 1 test. It means that un is absolutely convergent.
n 1 b) If L>1 then |un+1|>|un| for sufficiently large n. Therefore, {un} does not tend to 0 when n tends to infinity. This follows that u n diverges. n 1 1 1 If L=1, we take and 2.
Both of them satisfy L=1, but the former diverges and the n 1 n n 1 n latter converges. ( 1)n1 2n 1 ( 1) n 2 n ( n 1)! 2 EXAMPLE: converges absolutely, since lim lim 0 <1. n 1 n! n ( 1)n 2n n n1 n! The following test can be proved by the same manner.7 The nth root (Cauchy) Test: 8 Nguyen Thieu Huy 3. Theorem on Absolutely Convergent Series Theorem 4.
(Rearrangement of Terms) The terms of an absolutely convergent series can be rearranged in any order, and all such rearranged series will converge to the same sum. However, if the terms of a conditionally convergent series are suitably rearranged, the resulting series may diverge or converge to any desired sum. (Sums, Differences, and Products) The sum, difference, and product of two absolutely convergent series is absolutely convergent. The operations can be performed as for finite series.
9 Lecture on Infinite Series and Differential Equations CHAPTER 2: INFINITE SEQUENCES AND SERIES OF FUNCTIONS We open this chapter with the thought that functions could be expressed in series form. Such representation is illustrated by Observe that until this section the sequences and series depended on one element, n. Now there is variation with respect to x as well. This complexity requires the introduction of a new concept called uniform convergence, which, in turn, is fundamental in exploring the continuity, differentiation, and integrability of series.
Basic Concepts of Sequences and Series of Functions 1.1 Definitions: is said to be convergent in [a, b] if the sequence of partial sums {Sn(x)}, n= 1,2,3,. In such case we write lim S n=S(x) n and call S(x) the sum of the series. These definitions can be modified to include other intervals besides [a, b], such as (a, b), and so on. The domain of convergence (absolute or uniform) of a series is the set of values of x for which the series of functions converges (absolutely or uniformly).
Suppose un(x) = xn/n and -1/2 ≤x≤ 1. Now, think of the constant function F(x) = 0 on this interval. For any ε> 0 and any x in the interval, there is N such that for all 10 Nguyen Thieu Huy n > N we have |un(x) - F(x)| < ε, i. Since the limit does not depend on x, the sequence is uniformly convergent.2 Special tests for uniform convergence of series 1.
