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Calculus and Differential Equations II
Calculus and Differential Equations II MATH 250 B Sequences and series Sequences and series Calculus and Differential Equations II Sequences A sequence is an infinite list of numbers, s1; s2;:::; sn;::: , indexed by integers. 1n Example 1: Find the first five terms of s = (−1)n , n 3 n ≥ 1. Example 2: Find a formula for sn, n ≥ 1, given that its first five terms are 0; 2; 6; 14; 30. Some sequences are defined recursively. For instance, sn = 2 sn−1 + 3, n > 1, with s1 = 1. If lim sn = L, where L is a number, we say that the sequence n!1 (sn) converges to L. If such a limit does not exist or if L = ±∞, one says that the sequence diverges. Sequences and series Calculus and Differential Equations II Sequences (continued) 2n Example 3: Does the sequence converge? 5n 1 Yes 2 No n 5 Example 4: Does the sequence + converge? 2 n 1 Yes 2 No sin(2n) Example 5: Does the sequence converge? n Remarks: 1 A convergent sequence is bounded, i.e. one can find two numbers M and N such that M < sn < N, for all n's. 2 If a sequence is bounded and monotone, then it converges. Sequences and series Calculus and Differential Equations II Series A series is a pair of sequences, (Sn) and (un) such that n X Sn = uk : k=1 A geometric series is of the form 2 3 n−1 k−1 Sn = a + ax + ax + ax + ··· + ax ; uk = ax 1 − xn One can show that if x 6= 1, S = a . -
1 Mean Value Theorem 1 1.1 Applications of the Mean Value Theorem
Seunghee Ye Ma 8: Week 5 Oct 28 Week 5 Summary In Section 1, we go over the Mean Value Theorem and its applications. In Section 2, we will recap what we have covered so far this term. Topics Page 1 Mean Value Theorem 1 1.1 Applications of the Mean Value Theorem . .1 2 Midterm Review 5 2.1 Proof Techniques . .5 2.2 Sequences . .6 2.3 Series . .7 2.4 Continuity and Differentiability of Functions . .9 1 Mean Value Theorem The Mean Value Theorem is the following result: Theorem 1.1 (Mean Value Theorem). Let f be a continuous function on [a; b], which is differentiable on (a; b). Then, there exists some value c 2 (a; b) such that f(b) − f(a) f 0(c) = b − a Intuitively, the Mean Value Theorem is quite trivial. Say we want to drive to San Francisco, which is 380 miles from Caltech according to Google Map. If we start driving at 8am and arrive at 12pm, we know that we were driving over the speed limit at least once during the drive. This is exactly what the Mean Value Theorem tells us. Since the distance travelled is a continuous function of time, we know that there is a point in time when our speed was ≥ 380=4 >>> speed limit. As we can see from this example, the Mean Value Theorem is usually not a tough theorem to understand. The tricky thing is realizing when you should try to use it. Roughly speaking, we use the Mean Value Theorem when we want to turn the information about a function into information about its derivative, or vice-versa. -
3.3 Convergence Tests for Infinite Series
3.3 Convergence Tests for Infinite Series 3.3.1 The integral test We may plot the sequence an in the Cartesian plane, with independent variable n and dependent variable a: n X The sum an can then be represented geometrically as the area of a collection of rectangles with n=1 height an and width 1. This geometric viewpoint suggests that we compare this sum to an integral. If an can be represented as a continuous function of n, for real numbers n, not just integers, and if the m X sequence an is decreasing, then an looks a bit like area under the curve a = a(n). n=1 In particular, m m+2 X Z m+1 X an > an dn > an n=1 n=1 n=2 For example, let us examine the first 10 terms of the harmonic series 10 X 1 1 1 1 1 1 1 1 1 1 = 1 + + + + + + + + + : n 2 3 4 5 6 7 8 9 10 1 1 1 If we draw the curve y = x (or a = n ) we see that 10 11 10 X 1 Z 11 dx X 1 X 1 1 > > = − 1 + : n x n n 11 1 1 2 1 (See Figure 1, copied from Wikipedia) Z 11 dx Now = ln(11) − ln(1) = ln(11) so 1 x 10 X 1 1 1 1 1 1 1 1 1 1 = 1 + + + + + + + + + > ln(11) n 2 3 4 5 6 7 8 9 10 1 and 1 1 1 1 1 1 1 1 1 1 1 + + + + + + + + + < ln(11) + (1 − ): 2 3 4 5 6 7 8 9 10 11 Z dx So we may bound our series, above and below, with some version of the integral : x If we allow the sum to turn into an infinite series, we turn the integral into an improper integral. -
Transactions American Mathematical Society
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY EDITED BY A. A. ALBERT OSCAR ZARISKI ANTONI ZYGMUND WITH THE COOPERATION OF RICHARD BRAUER NELSON DUNFORD WILLIAM FELLER G. A. HEDLUND NATHAN JACOBSON IRVING KAPLANSKY S. C. KLEENE M. S. KNEBELMAN SAUNDERS MacLANE C. B. MORREY W. T. REID O. F. G. SCHILLING N. E. STEENROD J. J. STOKER D. J. STRUIK HASSLER WHITNEY R. L. WILDER VOLUME 62 JULY TO DECEMBER 1947 PUBLISHED BY THE SOCIETY MENASHA, WIS., AND NEW YORK 1947 Reprinted with the permission of The American Mathematical Society Johnson Reprint Corporation Johnson Reprint Company Limited 111 Fifth Avenue, New York, N. Y. 10003 Berkeley Square House, London, W. 1 First reprinting, 1964, Johnson Reprint Corporation PRINTED IN THE UNITED STATES OF AMERICA TABLE OF CONTENTS VOLUME 62, JULY TO DECEMBER, 1947 Arens, R. F., and Kelley, J. L. Characterizations of the space of con- tinuous functions over a compact Hausdorff space. 499 Baer, R. Direct decompositions. 62 Bellman, R. On the boundedness of solutions of nonlinear differential and difference equations. 357 Bergman, S. Two-dimensional subsonic flows of a compressible fluid and their singularities. 452 Blumenthal, L. M. Congruence and superposability in elliptic space.. 431 Chang, S. C. Errata for Contributions to projective theory of singular points of space curves. 548 Day, M. M. Polygons circumscribed about closed convex curves. 315 Day, M. M. Some characterizations of inner-product spaces. 320 Dushnik, B. Maximal sums of ordinals. 240 Eilenberg, S. Errata for Homology of spaces with operators. 1. 548 Erdös, P., and Fried, H. On the connection between gaps in power series and the roots of their partial sums. -
What Is on Today 1 Alternating Series
MA 124 (Calculus II) Lecture 17: March 28, 2019 Section A3 Professor Jennifer Balakrishnan, [email protected] What is on today 1 Alternating series1 1 Alternating series Briggs-Cochran-Gillett x8:6 pp. 649 - 656 We begin by reviewing the Alternating Series Test: P k+1 Theorem 1 (Alternating Series Test). The alternating series (−1) ak converges if 1. the terms of the series are nonincreasing in magnitude (0 < ak+1 ≤ ak, for k greater than some index N) and 2. limk!1 ak = 0. For series of positive terms, limk!1 ak = 0 does NOT imply convergence. For alter- nating series with nonincreasing terms, limk!1 ak = 0 DOES imply convergence. Example 2 (x8.6 Ex 16, 20, 24). Determine whether the following series converge. P1 (−1)k 1. k=0 k2+10 P1 1 k 2. k=0 − 5 P1 (−1)k 3. k=2 k ln2 k 1 MA 124 (Calculus II) Lecture 17: March 28, 2019 Section A3 Recall that if a series converges to a value S, then the remainder is Rn = S − Sn, where Sn is the sum of the first n terms of the series. An upper bound on the magnitude of the remainder (the absolute error) in an alternating series arises form the following observation: when the terms are nonincreasing in magnitude, the value of the series is always trapped between successive terms of the sequence of partial sums. Thus we have jRnj = jS − Snj ≤ jSn+1 − Snj = an+1: This justifies the following theorem: P1 k+1 Theorem 3 (Remainder in Alternating Series). -
Series: Convergence and Divergence Comparison Tests
Series: Convergence and Divergence Here is a compilation of what we have done so far (up to the end of October) in terms of convergence and divergence. • Series that we know about: P∞ n Geometric Series: A geometric series is a series of the form n=0 ar . The series converges if |r| < 1 and 1 a1 diverges otherwise . If |r| < 1, the sum of the entire series is 1−r where a is the first term of the series and r is the common ratio. P∞ 1 2 p-Series Test: The series n=1 np converges if p1 and diverges otherwise . P∞ • Nth Term Test for Divergence: If limn→∞ an 6= 0, then the series n=1 an diverges. Note: If limn→∞ an = 0 we know nothing. It is possible that the series converges but it is possible that the series diverges. Comparison Tests: P∞ • Direct Comparison Test: If a series n=1 an has all positive terms, and all of its terms are eventually bigger than those in a series that is known to be divergent, then it is also divergent. The reverse is also true–if all the terms are eventually smaller than those of some convergent series, then the series is convergent. P P P That is, if an, bn and cn are all series with positive terms and an ≤ bn ≤ cn for all n sufficiently large, then P P if cn converges, then bn does as well P P if an diverges, then bn does as well. (This is a good test to use with rational functions. -
J´Ozef Marcinkiewicz
JOZEF¶ MARCINKIEWICZ: ANALYSIS AND PROBABILITY N. H. BINGHAM, Imperial College London Pozna¶n,30 June 2010 JOZEF¶ MARCINKIEWICZ Life Born 4 March 1910, Cimoszka, Bialystok, Poland Student, 1930-33, University of Stefan Batory in Wilno (professors Stefan Kempisty, Juliusz Rudnicki and Antoni Zygmund) 1931-32: taught Lebesgue integration and trigono- metric series by Zygmund MA 1933; military service 1933-34 PhD 1935, under Zygmund 1935-36, Fellowship, U. Lw¶ow,with Kaczmarz and Schauder 1936, senior assistant, Wilno; dozent, 1937 Spring 1939, Fellowship, Paris; o®ered chair by U. Pozna¶n August 1939: in England; returned to Poland in anticipation of war (he was an o±cer in the reserve); already in uniform by 2 September Second lieutenant, 2nd Battalion, 205th In- fantry Regiment Defence of Lwo¶w12 - 21 September 1939; Lwo¶wsurrendered to Red (Soviet) Army Prisoner of war 25 September ("temporary in- ternment" by USSR); taken to Starobielsk Presumed executed Starobielsk, or Kharkov, or Kozielsk, or Katy¶n;Katy¶nMassacre commem- orated on 10 April Work We outline (most of) the main areas in which M's influence is directly seen today, and sketch the current state of (most of) his areas of in- terest { all in a very healthy state, an indication of M's (and Z's) excellent mathematical taste. 55 papers 1933-45 (the last few posthumous) Collaborators: Zygmund 15, S. Bergman 2, B. Jessen, S. Kaczmarz, R. Salem Papers (analysed by Zygmund) on: Functions of a real variable Trigonometric series Trigonometric interpolation Functional operations Orthogonal systems Functions of a complex variable Calculus of probability MATHEMATICS IN POLAND BETWEEN THE WARS K. -
Marshall Stone Y Las Matemáticas
MARSHALL STONE Y LAS MATEMÁTICAS Referencia: 1998. Semanario Universidad, San José. Costa Rica. Marshall Harvey Stone nació en Nueva York el 8 de abril de 1903. A los 16 años entró a Harvard y se graduó summa cum laude en 1922. Antes de ser profesor de Harvard entre 1933 y 1946, fue profesor en Columbia (1925-1927), Harvard (1929-1931), Yale (1931-1933) y Stanford en el verano de 1933. Aunque graduado y profesor en Harvard University, se conoce más por haber convertido el Departamento de Matemática de la University of Chicago -como su Director- en uno de los principales centros matemáticos del mundo, lo que logró con la contratación de los famosos matemáticos Andre Weil, S. S. Chern, Antoni Zygmund, Saunders Mac Lane y Adrian Albert. También fueron contratados en esa época: Paul Halmos, Irving Seal y Edwin Spanier. Para Saunders Mac Lane, el Departamento de Matemática que constituyó Stone en Chicago fue en su momento “sin duda el departamento de matemáticas líder en el país”., y probablemente, deberíamos añadir, en el mundo. Los méritos científicos de Stone fueron muchos. Cuando llegó a Chicago en 1946, por recomendación de John von Neumann al presidente de la Universidad de Chicago, ya había realizado importantes trabajos en varias áreas matemáticas, por ejemplo: la teoría espectral de operadores autoadjuntos en espacios de Hilbert y en las propiedades algebraicas de álgebras booleanas en el estudio de anillos de funciones continuas. Se le conoce por el famoso teorema de Stone Weierstrass, así como la compactificación de Stone Cech. Su libro más influyente fue Linear Transformations in Hilbert Space and their Application to Analysis. -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
Irving Kaplansky
Portraying and remembering Irving Kaplansky Hyman Bass University of Michigan Mathematical Sciences Research Institute • February 23, 2007 1 Irving (“Kap”) Kaplansky “infinitely algebraic” “I liked the algebraic way of looking at things. I’m additionally fascinated when the algebraic method is applied to infinite objects.” 1917 - 2006 A Gallery of Portraits 2 Family portrait: Kap as son • Born 22 March, 1917 in Toronto, (youngest of 4 children) shortly after his parents emigrated to Canada from Poland. • Father Samuel: Studied to be a rabbi in Poland; worked as a tailor in Toronto. • Mother Anna: Little schooling, but enterprising: “Health Bread Bakeries” supported (& employed) the whole family 3 Kap’s father’s grandfather Kap’s father’s parents Kap (age 4) with family 4 Family Portrait: Kap as father • 1951: Married Chellie Brenner, a grad student at Harvard Warm hearted, ebullient, outwardly emotional (unlike Kap) • Three children: Steven, Alex, Lucy "He taught me and my brothers a lot, (including) what is really the most important lesson: to do the thing you love and not worry about making money." • Died 25 June, 2006, at Steven’s home in Sherman Oaks, CA Eight months before his death he was still doing mathematics. Steven asked, -“What are you working on, Dad?” -“It would take too long to explain.” 5 Kap & Chellie marry 1951 Family portrait, 1972 Alex Steven Lucy Kap Chellie 6 Kap – The perfect accompanist “At age 4, I was taken to a Yiddish musical, Die Goldene Kala. It was a revelation to me that there could be this kind of entertainment with music. -
Mathematicians Fleeing from Nazi Germany
Mathematicians Fleeing from Nazi Germany Mathematicians Fleeing from Nazi Germany Individual Fates and Global Impact Reinhard Siegmund-Schultze princeton university press princeton and oxford Copyright 2009 © by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Siegmund-Schultze, R. (Reinhard) Mathematicians fleeing from Nazi Germany: individual fates and global impact / Reinhard Siegmund-Schultze. p. cm. Includes bibliographical references and index. ISBN 978-0-691-12593-0 (cloth) — ISBN 978-0-691-14041-4 (pbk.) 1. Mathematicians—Germany—History—20th century. 2. Mathematicians— United States—History—20th century. 3. Mathematicians—Germany—Biography. 4. Mathematicians—United States—Biography. 5. World War, 1939–1945— Refuges—Germany. 6. Germany—Emigration and immigration—History—1933–1945. 7. Germans—United States—History—20th century. 8. Immigrants—United States—History—20th century. 9. Mathematics—Germany—History—20th century. 10. Mathematics—United States—History—20th century. I. Title. QA27.G4S53 2008 510.09'04—dc22 2008048855 British Library Cataloging-in-Publication Data is available This book has been composed in Sabon Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10 987654321 Contents List of Figures and Tables xiii Preface xvii Chapter 1 The Terms “German-Speaking Mathematician,” “Forced,” and“Voluntary Emigration” 1 Chapter 2 The Notion of “Mathematician” Plus Quantitative Figures on Persecution 13 Chapter 3 Early Emigration 30 3.1. The Push-Factor 32 3.2. The Pull-Factor 36 3.D. -
Math 6B Notes Written by Victoria Kala [email protected] SH 6432U Office Hours: R 12:30 − 1:30Pm Last Updated 5/31/2016
Math 6B Notes Written by Victoria Kala [email protected] SH 6432u Office Hours: R 12:30 − 1:30pm Last updated 5/31/2016 Green's Theorem We say that a closed curve is positively oriented if it is oriented counterclockwise. It is negatively oriented if it is oriented clockwise. Theorem (Green's Theorem). Let C be a positively oriented, piecewise smooth, simple closed curve in the plane and let D be the region bounded by C. If F(x; y) = (P (x; y);Q(x; y)) with P and Q having continuous partial derivatives on an open region that contains D, then Z Z ZZ @Q @P F · ds = P dx + Qdy = − dA: C C D @x @y In other words, Green's Theorem allows us to change from a complicated line integral over a curve C to a less complicated double integral over the region bounded by C. The Operator r We define the vector differential operator r (called \del") as @ @ @ r = i + j + k: @x @y @z The gradient of a function f is given by @f @f @f rf = i + j + k: @x @y @z The curl of a function F = P i + Qj + Rk is defined as curl F = r × F: Written out more explicitly: i j k @ @ @ curl F = r × F = @x @y @z PQR @ @ @ @ @ @ = @y @z i − @x @z j + @x @y k QR PR PQ @R @Q @P @R @Q @P = − i + − j + − k @y @z @z @x @x @y 1 The divergence of function F = P i + Qj + Rk is defined as @P @Q @R div F = r · F = + + : @x @y @z Notice that the curl of a function is a vector and the divergence of a function is a scalar (just a number).