DOCSLIB.ORG

Series in Banach Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Series in Banach Spaces Operator Theory Advances and Applications Vol. 94 Editor I. Gohberg Editorial Office: T. Kailath (Stanford) School of Mathematical H.G. Kaper (Argonne) Sciences S.T. Kuroda (Tokyo) Tel Aviv University P. Lancaster (Calgary) Ramat Aviv, Israel L.E. Lerer (Haifa) E. Meister (Darmstadt) Editorial Board: B. Mityagin (Columbus) J. Arazy (Haifa) V.V. Peller (Manhattan, Kansas) A. Atzmon (Tel Aviv) J.D. Pincus (Stony Brook) J.A. Ball (Blackburg) M. Rosenblum (Charlottesville) A. Ben-Artzi (Tel Aviv) J. Rovnyak (Charlottesville) H. Bercovici (Bloomington) D.E. Sarason (Berkeley) A. Bottcher (Chemnitz) H. Upmeier (Marburg) L. de Branges (West Lafayette) S. M. Verduyn-Lunel (Amsterdam) K. Clancey (Athens, USA) D. Voiculescu (Berkeley) L.A. Coburn (Buffalo) H. Widom (Santa Cruz) K. R. Davidson (Waterloo, Ontario) D. Xia (Nashville) R. G. Douglas (Stony Brook) D. Yafaev (Rennes) H. Dym (Rehovot) A. Dynin (Columbus) P.A. Fillmore (Halifax) Honorary and Advisory C. Foias (Bloomington) Editorial Board: P.A. Fuhrmann (Beer Sheva) P.R. Halmos (Santa Clara) S. Goldberg (College Park) T. Kato (Berkeley) B. Gramsch (Mainz) P.O. Lax (New York) G. Heinig (Chemnitz) M.S. Livsic (Beer Sheva) J.A. Helton (La Jolla) R. Phillips (Stanford) M.A. Kaashoek (Amsterdam) B. Sz.-Nagy (Szeged) Series in Banach Spaces Conditional and Unconditional Convergence Mikhail I. Kadets Vladimir M. Kadets Translated from the Russian by Andrei lacob Birkhauser Verlag Basel . Boston . Berlin Authors M.l. Kadets and V.M. Kadets Pr. Pravdy 5, apt. 26 Kharkov 310022 Ukraine 1991 Mathematics Subject Classification 46B 15, 46B20, 46010, 42C20, 52A20, IOE05 , 60B 12 A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Deutsche Bibliothek Cataloging-in-Publication Data Kadec, Michail I.: Series in Banach spaces: conditional and unconditional convergence I Mikhail I. Kadets ; Vladimir M. Kadets. Trans!. from the Russ. by Andrei Iacob. - Basel; Boston; Berlin: Birkhauser, 1997 (Operator theory; Vo!. 94) ISBN-13: 978-3-0348-9942-0 e-ISBN-13: 978-3-0348-9196-7 DOl: 10.1007/978-3-0348-9196-7 NE: Kadec, Vladimir M.:; OT This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use the permission of the copyright holder must be obtained. © 1997 Birkhauser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Softcover reprint of the hardcover I st edition 1997 Printed on acid-free paper produced from chlorine-free pulp. TCF 00 Cover design: Heinz Hiltbrunner, Basel ISBN-13: 978-3-0348-9942-0 987654321 CONTENTS Introduction ... .... .. .... .... .. ....... ...... ..... .. .. .. ... vii Notations ..... .. .. .. ... .. ........... ..... .. ................. 1 Chapter 1. Background Material §l. Numerical Series. Riemann's Theorem....... .... ........ 5 §2. Main Definitions. Elementary Properties of Vector Series .. .. .. .. ... .. ....... .... ............ .... 7 §3. Preliminary Material on Rearrangements of Series of Elements of a Banach Space .. ... ... .... .. .. .... ... ... .. 9 Chapter 2. Series in a Finite-Dimensional Space §l. Steinitz's Theorem on the Sum Range of a Series . ........ 13 §2. The Dvoretzky-Hanani Theorem on Perfectly Divergent Series ...... ....... ..... ... .. ..... .. .. .. .. .. .. 21 §3. Pecherskii's Theorem . 23 Chapter 3. Conditional Convergence in an Infinite- Dimensional Space §l. Basic Counterexamples . 29 §2. A Series Whose Sum Range Consists of Two Points ... .... 32 §3. Chobanyan's Theorem ..... .. .... .. ........ ..... .. ....... 36 §4. The Khinchin Inequalities and the Theorem of M. I. Kadets on Conditionally Convergent Series in Lp . ... .. ... .. .... 39 Chapter 4. Unconditionally Convergent Series §l. The Dvoretzky-Rogers Theorem..... .. .. ..... .. ... .. ... 45 §2. Orlicz's Theorem on Unconditionally Convergent Series in Lp Spaces . .... .......... ... ..... ..... .. .. ... ... .. ... .. 49 §3. Absolutely Summing Operators. Grothendieck's Theorem 52 Chapter 5. Orlicz's Theorem and the Structure of Finite-Dimensional Subspaces §l. Finite Representability . ..... ... .. .. .. ... ...... .. ... .. .. 59 §2. The space Co, C-Convexity, and Orlicz's Theorem....... 62 §3. Survey on Results on Type and Cotype . ... ... .. .. .... .. 67 v vi CONTENTS Chapter 6. Some Results from the General Theory of Banach Spaces §1. Frechet Differentiability of Convex Functions ............ 71 §2. Dvoretzky's Theorem ................................. .. 73 §3. Basic Sequences .......................................... 79 §4. Some Applications to Conditionally Convergent Series 82 Chapter 7. Steinitz's Theorem and B-Convexity §1. Conditionally Convergent Series in Spaces with Infratype ............................ .... ........... 87 §2. A Technique for Transferring Examples with Nonlinear Sum Range to Arbitrary Infinite-Dimensional Banach Spaces. 93 §3. Series in Spaces That Are Not B-Convex ........... .. ... 97 Chapter 8. Rearrangements of Series in Topological Vector Spaces §1. Weak and Strong Sum Range ............................ 101 §2. Rearrangements of Series of Functions ..... .............. 106 §3. Banaszczyk's Theorem on Series in Metrizable Nuclear Spaces ........................................... 110 Appendix. The Limit Set of the Riemann Integral Sums of a Vector-Valued Function §1. Functions Valued in a B-Convex Space .. ................ 120 §2. The Example of Nakamura and Amemiya .. .............. 122 §3. Separability of the Space and the Structure of I(f) ........ 127 §4. Connection with the Weak Topology ..................... 131 Comments to the Exercises . .. 141 References................ ... ............................ .... 149 Index .. ..... ................................................ 155 INTRODUCTION Series of scalars, vectors, or functions are among the fundamental objects of mathematical analysis. When the arrangement of the terms is fixed, investigating a series amounts to investigating the sequence of its partial sums. In this case the theory of series is a part of the theory of sequences, which deals with their convergence, asymptotic behavior, etc. The specific character of the theory of series manifests itself when one considers rearrangements (permutations) of the terms of a series, which brings combinatorial considerations into the problems studied. The phenomenon that a numerical series can change its sum when the order of its terms is changed is one of the most impressive facts encountered in a university analysis course. The present book is devoted precisely to this aspect of the theory of series whose terms are elements of Banach (as well as other topological linear) spaces. The exposition focuses on two complementary problems. The first is to char­ acterize those series in a given space that remain convergent (and have the same sum) for any rearrangement of their terms; such series are usually called uncon­ ditionally convergent. The second problem is , when a series converges only for certain rearrangements of its terms (in other words, converges conditionally), to describe its sum range, i.e., the set of sums of all its convergent rearrangements. Recall that for series of real numbers both problems are settled by a well­ known theorem of Riemann (1867): a series converges unconditionally if and only if it converges absolutely; if the series converges conditionally, then its sum range is the whole real line. Difficultiet> arit>e when one considers more general series, for instance, se­ ries in a finite-dimensional space, or even series of complex numbers. To be more precise, the answer to the first problem is readily obtained, as before: in any finite-dimensional vector space the commutativity law (invariance under rearrange­ ments) holds only for absolutely convergent series. By contrast, the solution to the second problem-the description of the set of elements to which a series can con­ verge for a suitable rearrangement of its terms-requires considerably more efforts. For series of complex numbers such a description was given by P. Levy in 1905 [50], while for case of series in finite-dimensional spaces it was provided by E. Steinitz [87]. The study of rearrangements of series in infinite-dimensional spaces was initi­ ated by W. Orlicz [63 ,64] and subsequently developed by many mathematicians in vii viii INTRODUCTION different countries, and continues to be an active field of research. However, until now this theme found almost no exposition in textbooks and monographs. Several years ago we undertook the task of partially filling this gap by writ­ ing a textbook Rearrangements of Series in Banach Spaces [43], intended for stu­ dents at Tartu University. However, the beautiful results of D. V. Pecherskii, S. A. Chobanyan, W. Banaszczyk, and others, which appeared after that textbook was published, changed in essential manner the state of the art of the theory of series, have led to the development of new approaches, and in some instances even to a new point of view on the subject. For this reason we have decided to address again, but on a more modern level, the same topics, the result being the present book. One of our goals was to bring the topics under consideration and the theory of Banach spaces to the attention of young mathematicians. This explains the large number of exercises, among which are elementary as well as very difficult
Load more

Recommended publications
  • Riemann Sums Fundamental Theorem of Calculus Indefinite
    Riemann Sums Partition P = {x0, x1, . , xn} of an interval [a, b]. ck ∈ [xk−1, xk] Pn R(f, P, a, b) = k=1 f(ck)∆xk As the widths ∆xk of the subintervals approach 0, the Riemann Sums hopefully approach a limit, the integral of f from a to b, written R b a f(x) dx. Fundamental Theorem of Calculus Theorem 1 (FTC-Part I). If f is continuous on [a, b], then F (x) = R x 0 a f(t) dt is defined on [a, b] and F (x) = f(x). Theorem 2 (FTC-Part II). If f is continuous on [a, b] and F (x) = R R b b f(x) dx on [a, b], then a f(x) dx = F (x) a = F (b) − F (a). Indefinite Integrals Indefinite Integral: R f(x) dx = F (x) if and only if F 0(x) = f(x). In other words, the terms indefinite integral and antiderivative are synonymous. Every differentiation formula yields an integration formula. Substitution Rule For Indefinite Integrals: If u = g(x), then R f(g(x))g0(x) dx = R f(u) du. For Definite Integrals: R b 0 R g(b) If u = g(x), then a f(g(x))g (x) dx = g(a) f( u) du. Steps in Mechanically Applying the Substitution Rule Note: The variables do not have to be called x and u. (1) Choose a substitution u = g(x). du (2) Calculate = g0(x). dx du (3) Treat as if it were a fraction, a quotient of differentials, and dx du solve for dx, obtaining dx = .
    [Show full text]
  • The Radius of Convergence of the Lam\'{E} Function
    The radius of convergence of the Lame´ function Yoon-Seok Choun Department of Physics, Hanyang University, Seoul, 133-791, South Korea Abstract We consider the radius of convergence of a Lame´ function, and we show why Poincare-Perron´ theorem is not applicable to the Lame´ equation. Keywords: Lame´ equation; 3-term recursive relation; Poincare-Perron´ theorem 2000 MSC: 33E05, 33E10, 34A25, 34A30 1. Introduction A sphere is a geometric perfect shape, a collection of points that are all at the same distance from the center in the three-dimensional space. In contrast, the ellipsoid is an imperfect shape, and all of the plane sections are elliptical or circular surfaces. The set of points is no longer the same distance from the center of the ellipsoid. As we all know, the nature is nonlinear and geometrically imperfect. For simplicity, we usually linearize such systems to take a step toward the future with good numerical approximation. Indeed, many geometric spherical objects are not perfectly spherical in nature. The shape of those objects are better interpreted by the ellipsoid because they rotates themselves. For instance, ellipsoidal harmonics are shown in the calculation of gravitational potential [11]. But generally spherical harmonics are preferable to mathematically complex ellipsoid harmonics. Lame´ functions (ellipsoidal harmonic fuctions) [7] are applicable to diverse areas such as boundary value problems in ellipsoidal geometry, Schrodinger¨ equations for various periodic and anharmonic potentials, the theory of Bose-Einstein condensates, group theory to quantum mechanical bound states and band structures of the Lame´ Hamiltonian, etc. As we apply a power series into the Lame´ equation in the algebraic form or Weierstrass’s form, the 3-term recurrence relation starts to arise and currently, there are no general analytic solutions in closed forms for the 3-term recursive relation of a series solution because of their mathematical complexity [1, 5, 13].
    [Show full text]
  • Absolute and Conditional Convergence, and Talk a Little Bit About the Mathematical Constant E
    MATH 8, SECTION 1, WEEK 3 - RECITATION NOTES TA: PADRAIC BARTLETT Abstract. These are the notes from Friday, Oct. 15th's lecture. In this talk, we study absolute and conditional convergence, and talk a little bit about the mathematical constant e. 1. Random Question Question 1.1. Suppose that fnkg is the sequence consisting of all natural numbers that don't have a 9 anywhere in their digits. Does the corresponding series 1 X 1 nk k=1 converge? 2. Absolute and Conditional Convergence: Definitions and Theorems For review's sake, we repeat the definitions of absolute and conditional conver- gence here: P1 P1 Definition 2.1. A series n=1 an converges absolutely iff the series n=1 janj P1 converges; it converges conditionally iff the series n=1 an converges but the P1 series of absolute values n=1 janj diverges. P1 (−1)n+1 Example 2.2. The alternating harmonic series n=1 n converges condition- ally. This follows from our work in earlier classes, where we proved that the series itself converged (by looking at partial sums;) conversely, the absolute values of this series is just the harmonic series, which we know to diverge. As we saw in class last time, most of our theorems aren't set up to deal with series whose terms alternate in sign, or even that have terms that occasionally switch sign. As a result, we developed the following pair of theorems to help us manipulate series with positive and negative terms: 1 Theorem 2.3. (Alternating Series Test / Leibniz's Theorem) If fangn=0 is a se- quence of positive numbers that monotonically decreases to 0, the series 1 X n (−1) an n=1 converges.
    [Show full text]
  • Dirichlet’S Test
    4. CONDITIONALLY CONVERGENT SERIES 23 4. Conditionally convergent series Here basic tests capable of detecting conditional convergence are studied. 4.1. Dirichlet’s test. Theorem 4.1. (Dirichlet’s test) n Let a complex sequence {An}, An = k=1 ak, be bounded, and the real sequence {bn} is decreasing monotonically, b1 ≥ b2 ≥ b3 ≥··· , so that P limn→∞ bn = 0. Then the series anbn converges. A proof is based on the identity:P m m m m−1 anbn = (An − An−1)bn = Anbn − Anbn+1 n=k n=k n=k n=k−1 X X X X m−1 = An(bn − bn+1)+ Akbk − Am−1bm n=k X Since {An} is bounded, there is a number M such that |An|≤ M for all n. Therefore it follows from the above identity and non-negativity and monotonicity of bn that m m−1 anbn ≤ An(bn − bn+1) + |Akbk| + |Am−1bm| n=k n=k X X m−1 ≤ M (bn − bn+1)+ bk + bm n=k ! X ≤ 2Mbk By the hypothesis, bk → 0 as k → ∞. Therefore the right side of the above inequality can be made arbitrary small for all sufficiently large k and m ≥ k. By the Cauchy criterion the series in question converges. This completes the proof. Example. By the root test, the series ∞ zn n n=1 X converges absolutely in the disk |z| < 1 in the complex plane. Let us investigate the convergence on the boundary of the disk. Put z = eiθ 24 1. THE THEORY OF CONVERGENCE ikθ so that ak = e and bn = 1/n → 0 monotonically as n →∞.
    [Show full text]
  • Chapter 3: Infinite Series
    Chapter 3: Infinite series Introduction Definition: Let (an : n ∈ N) be a sequence. For each n ∈ N, we define the nth partial sum of (an : n ∈ N) to be: sn := a1 + a2 + . an. By the series generated by (an : n ∈ N) we mean the sequence (sn : n ∈ N) of partial sums of (an : n ∈ N) (ie: a series is a sequence). We say that the series (sn : n ∈ N) generated by a sequence (an : n ∈ N) is convergent if the sequence (sn : n ∈ N) of partial sums converges. If the series (sn : n ∈ N) is convergent then we call its limit the sum of the series and we write: P∞ lim sn = an. In detail: n→∞ n=1 s1 = a1 s2 = a1 + a + 2 = s1 + a2 s3 = a + 1 + a + 2 + a3 = s2 + a3 ... = ... sn = a + 1 + ... + an = sn−1 + an. Definition: A series that is not convergent is called divergent, ie: each series is ei- P∞ ther convergent or divergent. It has become traditional to use the notation n=1 an to represent both the series (sn : n ∈ N) generated by the sequence (an : n ∈ N) and the sum limn→∞ sn. However, this ambiguity in the notation should not lead to any confu- sion, provided that it is always made clear that the convergence of the series must be established. Important: Although traditionally series have been (and still are) represented by expres- P∞ sions such as n=1 an they are, technically, sequences of partial sums. So if somebody comes up to you in a supermarket and asks you what a series is - you answer “it is a P∞ sequence of partial sums” not “it is an expression of the form: n=1 an.” n−1 Example: (1) For 0 < r < ∞, we may define the sequence (an : n ∈ N) by, an := r for each n ∈ N.
    [Show full text]
  • Chapter 11 Outline Math 236 Summer 2002
    Chapter 11 Outline Math 236 Summer 2002 11.1 Infinite Series Sigma notation: Be able to go from a written-out sum to sigma notation and back. • Know how to change indicies and algebraically manipulate a sum in sigma notation. Infinite series as sequences of partial sums: What does it mean to say that an • infinite series converges? See Definition 11.1.1 which describes a series as the limit of its sequence of partial sums. Do not confuse the series with the corresponding sequence! Telescoping series: Sometimes the terms of a series can be rewritten using partial • fractions so that you can find a formula for the nth partial sum for the series. Then you can take the limit of this sequence of partial sums to get the sum of the series (if it exists). Be able to do this and be able to explain what you are doing (including notation) while you do it. Not all series are \telescoping" like this. The geometric series: What is the general form of a geometric series? When • does it converge, and to what? Notice that the formula for the sum of a convergent geometric series only works if you start from k = 0. Be able to prove (in a particular 1 case like x = 4 or in general) the formula for the sum of a convergent geometric series. The proof involves finding a formula for the nth partial sum of the series (start by computing sn xsn) and then taking the limit of that formula to get the sum of the series.
    [Show full text]
  • 2.53 Means the Reference Is to Be Found in Question 53 of Chapter 2
    Cambridge University Press 978-1-107-44453-9 - Numbers and Functions: Steps into Analysis: Third Edition R. P. Burn Index More information Index 2.53 means the reference is to be found in question 53 of chapter 2. 2.53− means that the reference is to be found before that question. 2.53+ means that the reference is to be found after that question. 2.53–58 refers to questions 53 to 58 of chapter 2. 2.53,64 refers to questions 53 and 64 of chapter 2. 2H means that the reference is to be found in the historical note to chapter 2. − Abel 5.91, 5H, 12.40 , 12H binomial theorem 1.5, 5.98, 112–113, 8H, absolute convergence 5.66–67, 77 11H, 12.45, 12H absolute value 2.52–64, 3.33, 54(vii) blancmange function 12.46 − accumulation point (cluster point) 4.48 Bolzano 2H, 4.53, 4H, 5H, 6.70, 6H, 7.12, alternating series test 5.62–63 7H, 12H + + + anti-derivative 10.52 , 54 Bolzano–Weierstrass theorem 4.46 , 53 + arc cosine 10.49–50 bound 3.48 , 3H − − Archimedean order 3.19 , 3.24, 3H, App. 1 lower: for sequence 3.5–7;forset4.53 , + Archimedes 1H, 2H, 3.25, 3H, 5H, 10H 7.8 − arc length 10.43–48 upper: for sequence 3.5–7;forset4.53 , + arc sine 8.44 7.8 arc tangent 1.8, 5.84, 8.45, 11.60–62 bounded sequence 3.5, 13–14, 62, 4.43–47, area 10.1–9 83–84, 5.25 under y = xk 10.3–5 convergent subsequence 3.80, 4.46–47, under y = x2 10.1 83–84 under y = x3 10.2 eventually 3.13, 62 under y = 1/x 10.8–9 monotonic 3.80, 4.33–35 arithmetic mean 2.30–39 Bouquet and Briot 5H Ascoli 4H Bourbaki 6H + associative law 5.1 , App.
    [Show full text]
  • 1 Convergence Tests
    Lecture 24Section 11.4 Absolute and Conditional Convergence; Alternating Series Jiwen He 1 Convergence Tests Basic Series that Converge or Diverge Basic Series that Converge X Geometric series: xk, if |x| < 1 X 1 p-series: , if p > 1 kp Basic Series that Diverge X Any series ak for which lim ak 6= 0 k→∞ X 1 p-series: , if p ≤ 1 kp Convergence Tests (1) Basic Test for Convergence P Keep in Mind that, if ak 9 0, then the series ak diverges; therefore there is no reason to apply any special convergence test. P k P k k Examples 1. x with |x| ≥ 1 (e.g, (−1) ) diverge since x 9 0. [1ex] k X k X 1 diverges since k → 1 6= 0. [1ex] 1 − diverges since k + 1 k+1 k 1 k −1 ak = 1 − k → e 6= 0. Convergence Tests (2) Comparison Tests Rational terms are most easily handled by basic comparison or limit comparison with p-series P 1/kp Basic Comparison Test 1 X 1 X 1 X k3 converges by comparison with converges 2k3 + 1 k3 k5 + 4k4 + 7 X 1 X 1 X 2 by comparison with converges by comparison with k2 k3 − k2 k3 X 1 X 1 X 1 diverges by comparison with diverges by 3k + 1 3(k + 1) ln(k + 6) X 1 comparison with k + 6 Limit Comparison Test X 1 X 1 X 3k2 + 2k + 1 converges by comparison with . diverges k3 − 1 √ k3 k3 + 1 X 3 X 5 k + 100 by comparison with √ √ converges by comparison with k 2k2 k − 9 k X 5 2k2 Convergence Tests (3) Root Test and Ratio Test The root test is used only if powers are involved.
    [Show full text]
  • When We Discuss Conditional Convergence, We Usually Note That
    When we discuss conditional convergence, we usually note that there is really only one way | namely, using the alternating series test | that you know of that can prove that a series converges when it does not converge absolutely. But not all signed series are strictly alternating! Here is a nice example which I claim converges conditionally, but not absolutely: X sin(n) n n=1 That's the sine of n radians there, meaning the terms in the series change sign often, but not by alternating strictly. Also the terms do not decrease monotonically in magnitude. The first few terms are approximately 0:84147 + 0:45464 + 0:04704 − 0:18920 − 0:19178 − 0:04656 + 0:09385 ::: Apart from the tests we discussed | the alternating series test and a few (e.g. integral test, comparison, ratio, etc.) which can only test positive series | the only hope you have of deciding convergence, in general, is to be able to find a supple formula for the partial sums. I will do that | sort of | in this example, showing a new idea. This \trick" is used pretty often, actually, as a way of speeding up convergence of a series, for those times when you have to know what a convergent series converges to. Let bn = sin(1) + sin(2) + ::: + sin(n), that is, the bn are partial sums for a different series (one without denominators). We're interested in these because of a little trick: with this definition of the bn we have b1 = sin(1) and for all other n > 1 we have sin(n) = bn − bn−1 = −bn−1 + bn We substitute this into our original series, hoping for a nice \formula" for the partial sums.
    [Show full text]
  • Math 1172 - Sequences and Series - Student Handout Modifications to Chapters 9 and 10
    Math 1172 - Sequences and Series - Student Handout Modifications to Chapters 9 and 10 Note: Due to the fact that Math 1172 does not cover certain topics in chapters 9 and 10, some of the exercise problems in the book will not be relevant. Some of the problems that you may skip are given below. If you are not sure about the relevance of a certain exercise, check with your instructor. Chapter 9 - Sequences and Infinite Series 9.1 The entire section is covered. 9.2 The entire section is covered except for the following: Not covered: Formal Definition of a Limit of a Sequence (bottom of page 635 and page 636, and problems 69-74 in the exercises.) 9.3 The entire section is covered. 9.4 Only some of this section is covered (see below.) Covered: The Divergence Test (pg. 648-649.) Includes THEOREM 9.8. Covered: The Harmonic Series (pg. 649-651.) Includes THEOREM 9.9. However the proof for the divergence of the Harmonic series will not involve using an integral as in the textbook on pages 650 and 651. Covered: Properties of Convergent Series (pg. 657-659.) Includes THEOREM 9.13. Not covered: The Integral Test (middle of page 651 to page 653, and problems 19-28 in the exercises.) Not covered: Convergence of p-Series (pg. 653-654, and problems 29-34 in the exercises.) Not covered: Estimating Series with Positive Terms (bottom of page 654 to top of page 657, and problems 35-42 in the exercises.) 9.5 Only one topic is covered.
    [Show full text]
  • 2. Infinite Series
    2. INFINITE SERIES 2.1. A PRE-REQUISITE:SEQUENCES We concluded the last section by asking what we would get if we considered the “Taylor polynomial of degree for the function ex centered at 0”, x2 x3 1 x 2! 3! As we said at the time, we have a lot of groundwork to consider first, such as the funda- mental question of what it even means to add an infinite list of numbers together. As we will see in the next section, this is a delicate question. In order to put our explorations on solid ground, we begin by studying sequences. A sequence is just an ordered list of objects. Our sequences are (almost) always lists of real numbers, so another definition for us would be that a sequence is a real-valued function whose domain is the positive integers. The sequence whose nth term is an is denoted an , or if there might be confusion otherwise, an n 1, which indicates that the sequence starts when n 1 and continues forever. Sequences are specified in several different ways. Perhaps the simplest way is to spec- ify the first few terms, for example an 2, 4, 6, 8, 10, 12, 14,... , is a perfectly clear definition of the sequence of positive even integers. This method is slightly less clear when an 2, 3, 5, 7, 11, 13, 17,... , although with a bit of imagination, one can deduce that an is the nth prime number (for technical reasons, 1 is not considered to be a prime number). Of course, this method com- pletely breaks down when the sequence has no discernible pattern, such as an 0, 4, 3, 2, 11, 29, 54, 59, 35, 41, 46,..
    [Show full text]
  • Comprehensive Problem Bank for MA 108 (Formerly Part of MA 203)
    Department of Mathematics Indian Institute of Technology, Bombay Powai, Mumbai{400 076, INDIA. MA 203{Mathematics III Autumn 2006 Instructors: A. Athavale N. Nataraj R. Raghunathan (Instructor in-charge) G. K. Srinivasan Name : Roll No : Course contents of MA 203 (Mathematics III): Ordinary differential equations of the 1st order, exactness and integrating factors, variation of pa- rameters, Picard's iteration method. Ordinary linear differential equations of the nth order, solution of homogeneous and non-homogeneous equations. The operator method. The methods ofundetermined co- efficients and variation of parameters. Systems of differential equations, Phase plane. Critical points, stability. Infinite sequences and series of real and complex numbers. Improper integrals. The Cauchy criterion, tests of convergence, absolute and conditional convergence. Series of functions. Improper integrals depend- ing on a parameter. Uniform convergence. Power series, radius of convergence. Power series methods for solutions of ordinary differential equations. Legendre equation and Legendre polynomials, Bessel equations and Bessel functions of first and second kind. Orthogonal sets of functions. Sturm-Liouville problems. Or- thogonality of Bessel functions and Legendre polynomials. The Laplace transform. The Inverse transform. Shifting properties, convolutions, partial fractions. Fourier series, half-range expansions. Approximation by trignometric polynomials. Fourier integrals. Texts/References E. Kreyszig, Advanced Engineering Mathematics, 8th ed., Wiley Eastern, 1999. Teaching Plan [K] refers to the text book by E. Kreyszig, \Advanced Engineering Mathematics", 8th Edition, John Wiley and Sons(1999). Policy for Attendance Attendance in both lectures and tutorial classes is compulsory. Students who fail to attend 80% of the lectures and tutorial classes may be awarded an XX grade. Evaluation: Figures in parentheses denote the percentage of the marks assigned to each quiz or exam.
    [Show full text]