Infinite Series
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Math 142 – Quiz 6 – Solutions 1. (A) by Direct Calculation, Lim 1+3N 2
Math 142 – Quiz 6 – Solutions 1. (a) By direct calculation, 1 + 3n 1 + 3 0 + 3 3 lim = lim n = = − . n→∞ n→∞ 2 2 − 5n n − 5 0 − 5 5 3 So it is convergent with limit − 5 . (b) By using f(x), where an = f(n), x2 2x 2 lim = lim = lim = 0. x→∞ ex x→∞ ex x→∞ ex (Obtained using L’Hˆopital’s Rule twice). Thus the sequence is convergent with limit 0. (c) By comparison, n − 1 n + cos(n) n − 1 ≤ ≤ 1 and lim = 1 n + 1 n + 1 n→∞ n + 1 n n+cos(n) o so by the Squeeze Theorem, the sequence n+1 converges to 1. 2. (a) By the Ratio Test (or as a geometric series), 32n+2 32n 3232n 7n 9 L = lim (16 )/(16 ) = lim = . n→∞ 7n+1 7n n→∞ 32n 7 7n 7 The ratio is bigger than 1, so the series is divergent. Can also show using the Divergence Test since limn→∞ an = ∞. (b) By the integral test Z ∞ Z t 1 1 t dx = lim dx = lim ln(ln x)|2 = lim ln(ln t) − ln(ln 2) = ∞. 2 x ln x t→∞ 2 x ln x t→∞ t→∞ So the integral diverges and thus by the Integral Test, the series also diverges. (c) By comparison, (n3 − 3n + 1)/(n5 + 2n3) n5 − 3n3 + n2 1 − 3n−2 + n−3 lim = lim = lim = 1. n→∞ 1/n2 n→∞ n5 + 2n3 n→∞ 1 + 2n−2 Since P 1/n2 converges (p-series, p = 2 > 1), by the Limit Comparison Test, the series converges. -
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. -
On a Series of Goldbach and Euler Llu´Is Bibiloni, Pelegr´I Viader, and Jaume Parad´Is
On a Series of Goldbach and Euler Llu´ıs Bibiloni, Pelegr´ı Viader, and Jaume Parad´ıs 1. INTRODUCTION. Euler’s paper Variae observationes circa series infinitas [6] ought to be considered important for several reasons. It contains the first printed ver- sion of Euler’s product for the Riemann zeta-function; it definitely establishes the use of the symbol π to denote the perimeter of the circle of diameter one; and it introduces a legion of interesting infinite products and series. The first of these is Theorem 1, which Euler says was communicated to him and proved by Goldbach in a letter (now lost): 1 = 1. n − m,n≥2 m 1 (One must avoid repetitions in this sum.) We refer to this result as the “Goldbach-Euler Theorem.” Goldbach and Euler’s proof is a typical example of what some historians consider a misuse of divergent series, for it starts by assigning a “value” to the harmonic series 1/n and proceeds by manipulating it by substraction and replacement of other series until the desired result is reached. This unchecked use of divergent series to obtain valid results was a standard procedure in the late seventeenth and early eighteenth centuries. It has provoked quite a lot of criticism, correction, and, why not, praise of the audacity of the mathematicians of the time. They were led by Euler, the “Master of Us All,” as Laplace christened him. We present the original proof of the Goldbach- Euler theorem in section 2. Euler was obviously familiar with other instances of proofs that used divergent se- ries. -
Series, Cont'd
Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 5 Notes These notes correspond to Section 8.2 in the text. Series, cont'd In the previous lecture, we defined the concept of an infinite series, and what it means for a series to converge to a finite sum, or to diverge. We also worked with one particular type of series, a geometric series, for which it is particularly easy to determine whether it converges, and to compute its limit when it does exist. Now, we consider other types of series and investigate their behavior. Telescoping Series Consider the series 1 X 1 1 − : n n + 1 n=1 If we write out the first few terms, we obtain 1 X 1 1 1 1 1 1 1 1 1 − = 1 − + − + − + − + ··· n n + 1 2 2 3 3 4 4 5 n=1 1 1 1 1 1 1 = 1 + − + − + − + ··· 2 2 3 3 4 4 = 1: We see that nearly all of the fractions cancel one another, revealing the limit. This is an example of a telescoping series. It turns out that many series have this property, even though it is not immediately obvious. Example The series 1 X 1 n(n + 2) n=1 is also a telescoping series. To see this, we compute the partial fraction decomposition of each term. This decomposition has the form 1 A B = + : n(n + 2) n n + 2 1 To compute A and B, we multipy both sides by the common denominator n(n + 2) and obtain 1 = A(n + 2) + Bn: Substituting n = 0 yields A = 1=2, and substituting n = −2 yields B = −1=2. -
1 Improper Integrals
July 14, 2019 MAT136 { Week 6 Justin Ko 1 Improper Integrals In this section, we will introduce the notion of integrals over intervals of infinite length or integrals of functions with an infinite discontinuity. These definite integrals are called improper integrals, and are understood as the limits of the integrals we introduced in Week 1. Definition 1. We define two types of improper integrals: 1. Infinite Region: If f is continuous on [a; 1) or (−∞; b], the integral over an infinite domain is defined as the respective limit of integrals over finite intervals, Z 1 Z t Z b Z b f(x) dx = lim f(x) dx; f(x) dx = lim f(x) dx: a t!1 a −∞ t→−∞ t R 1 R a If both a f(x) dx < 1 and −∞ f(x) dx < 1, then Z 1 Z a Z 1 f(x) dx = f(x) dx + f(x) dx: −∞ −∞ a R 1 If one of the limits do not exist or is infinite, then −∞ f(x) dx diverges. 2. Infinite Discontinuity: If f is continuous on [a; b) or (a; b], the improper integral for a discon- tinuous function is defined as the respective limit of integrals over finite intervals, Z b Z t Z b Z b f(x) dx = lim f(x) dx f(x) dx = lim f(x) dx: a t!b− a a t!a+ t R c R b If f has a discontinuity at c 2 (a; b) and both a f(x) dx < 1 and c f(x) dx < 1, then Z b Z c Z b f(x) dx = f(x) dx + f(x) dx: a a c R b If one of the limits do not exist or is infinite, then a f(x) dx diverges. -
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. -
Math 137 Calculus 1 for Honours Mathematics Course Notes
Math 137 Calculus 1 for Honours Mathematics Course Notes Barbara A. Forrest and Brian E. Forrest Version 1.61 Copyright c Barbara A. Forrest and Brian E. Forrest. All rights reserved. August 1, 2021 All rights, including copyright and images in the content of these course notes, are owned by the course authors Barbara Forrest and Brian Forrest. By accessing these course notes, you agree that you may only use the content for your own personal, non-commercial use. You are not permitted to copy, transmit, adapt, or change in any way the content of these course notes for any other purpose whatsoever without the prior written permission of the course authors. Author Contact Information: Barbara Forrest ([email protected]) Brian Forrest ([email protected]) i QUICK REFERENCE PAGE 1 Right Angle Trigonometry opposite ad jacent opposite sin θ = hypotenuse cos θ = hypotenuse tan θ = ad jacent 1 1 1 csc θ = sin θ sec θ = cos θ cot θ = tan θ Radians Definition of Sine and Cosine The angle θ in For any θ, cos θ and sin θ are radians equals the defined to be the x− and y− length of the directed coordinates of the point P on the arc BP, taken positive unit circle such that the radius counter-clockwise and OP makes an angle of θ radians negative clockwise. with the positive x− axis. Thus Thus, π radians = 180◦ sin θ = AP, and cos θ = OA. 180 or 1 rad = π . The Unit Circle ii QUICK REFERENCE PAGE 2 Trigonometric Identities Pythagorean cos2 θ + sin2 θ = 1 Identity Range −1 ≤ cos θ ≤ 1 −1 ≤ sin θ ≤ 1 Periodicity cos(θ ± 2π) = cos θ sin(θ ± 2π) = sin -
G-Convergence and Homogenization of Some Sequences of Monotone Differential Operators
Thesis for the degree of Doctor of Philosophy Östersund 2009 G-CONVERGENCE AND HOMOGENIZATION OF SOME SEQUENCES OF MONOTONE DIFFERENTIAL OPERATORS Liselott Flodén Supervisors: Associate Professor Anders Holmbom, Mid Sweden University Professor Nils Svanstedt, Göteborg University Professor Mårten Gulliksson, Mid Sweden University Department of Engineering and Sustainable Development Mid Sweden University, SE‐831 25 Östersund, Sweden ISSN 1652‐893X, Mid Sweden University Doctoral Thesis 70 ISBN 978‐91‐86073‐36‐7 i Akademisk avhandling som med tillstånd av Mittuniversitetet framläggs till offentlig granskning för avläggande av filosofie doktorsexamen onsdagen den 3 juni 2009, klockan 10.00 i sal Q221, Mittuniversitetet, Östersund. Seminariet kommer att hållas på svenska. G-CONVERGENCE AND HOMOGENIZATION OF SOME SEQUENCES OF MONOTONE DIFFERENTIAL OPERATORS Liselott Flodén © Liselott Flodén, 2009 Department of Engineering and Sustainable Development Mid Sweden University, SE‐831 25 Östersund Sweden Telephone: +46 (0)771‐97 50 00 Printed by Kopieringen Mittuniversitetet, Sundsvall, Sweden, 2009 ii Tothememoryofmyfather G-convergence and Homogenization of some Sequences of Monotone Differential Operators Liselott Flodén Department of Engineering and Sustainable Development Mid Sweden University, SE-831 25 Östersund, Sweden Abstract This thesis mainly deals with questions concerning the convergence of some sequences of elliptic and parabolic linear and non-linear oper- ators by means of G-convergence and homogenization. In particular, we study operators with oscillations in several spatial and temporal scales. Our main tools are multiscale techniques, developed from the method of two-scale convergence and adapted to the problems stud- ied. For certain classes of parabolic equations we distinguish different cases of homogenization for different relations between the frequen- cies of oscillations in space and time by means of different sets of local problems. -
Generalizations of the Riemann Integral: an Investigation of the Henstock Integral
Generalizations of the Riemann Integral: An Investigation of the Henstock Integral Jonathan Wells May 15, 2011 Abstract The Henstock integral, a generalization of the Riemann integral that makes use of the δ-fine tagged partition, is studied. We first consider Lebesgue’s Criterion for Riemann Integrability, which states that a func- tion is Riemann integrable if and only if it is bounded and continuous almost everywhere, before investigating several theoretical shortcomings of the Riemann integral. Despite the inverse relationship between integra- tion and differentiation given by the Fundamental Theorem of Calculus, we find that not every derivative is Riemann integrable. We also find that the strong condition of uniform convergence must be applied to guarantee that the limit of a sequence of Riemann integrable functions remains in- tegrable. However, by slightly altering the way that tagged partitions are formed, we are able to construct a definition for the integral that allows for the integration of a much wider class of functions. We investigate sev- eral properties of this generalized Riemann integral. We also demonstrate that every derivative is Henstock integrable, and that the much looser requirements of the Monotone Convergence Theorem guarantee that the limit of a sequence of Henstock integrable functions is integrable. This paper is written without the use of Lebesgue measure theory. Acknowledgements I would like to thank Professor Patrick Keef and Professor Russell Gordon for their advice and guidance through this project. I would also like to acknowledge Kathryn Barich and Kailey Bolles for their assistance in the editing process. Introduction As the workhorse of modern analysis, the integral is without question one of the most familiar pieces of the calculus sequence. -
Calculus II Chapter 9 Review Name______
Calculus II Chapter 9 Review Name___________ ________________________ SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. an 1) a = 1, a = 1) 1 n+1 n + 2 Find a formula for the nth term of the sequence. 1 1 1 1 2) 1, - , , - , 2) 4 9 16 25 Find the limit of the sequence if it converges; otherwise indicate divergence. 9 + 8n 3) a = 3) n 9 + 5n 3 4) (-1)n 1 - 4) n Determine if the sequence is bounded. Δn 5) 5) 4n Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. 3 3 3 3 6) 3 - + - + ... + (-1)n-1 + ... 6) 8 64 512 8n-1 7 7 7 7 7) + + + ... + + ... 7) 1·3 2·4 3·5 n(n + 2) Find the sum of the series. Q n 9 8) _ (-1) 8) 4n n=0 Use partial fractions to find the sum of the series. 3 9) 9) _ (4n - 1)(4n + 3) 1 Determine if the series converges or diverges; if the series converges, find its sum. 3n+1 10) _ 10) 7n-1 1 cos nΔ 11) _ 11) 7n Find the values of x for which the geometric series converges. n n 12) _ -3 x 12) Find the sum of the geometric series for those x for which the series converges. -
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 -
3.2 Introduction to Infinite Series
3.2 Introduction to Infinite Series Many of our infinite sequences, for the remainder of the course, will be defined by sums. For example, the sequence m X 1 S := : (1) m 2n n=1 is defined by a sum. Its terms (partial sums) are 1 ; 2 1 1 3 + = ; 2 4 4 1 1 1 7 + + = ; 2 4 8 8 1 1 1 1 15 + + + = ; 2 4 8 16 16 ::: These infinite sequences defined by sums are called infinite series. Review of sigma notation The Greek letter Σ used in this notation indicates that we are adding (\summing") elements of a certain pattern. (We used this notation back in Calculus 1, when we first looked at integrals.) Here our sums may be “infinite”; when this occurs, we are really looking at a limit. Resources An introduction to sequences a standard part of single variable calculus. It is covered in every calculus textbook. For example, one might look at * section 11.3 (Integral test), 11.4, (Comparison tests) , 11.5 (Ratio & Root tests), 11.6 (Alternating, abs. conv & cond. conv) in Calculus, Early Transcendentals (11th ed., 2006) by Thomas, Weir, Hass, Giordano (Pearson) * section 11.3 (Integral test), 11.4, (comparison tests), 11.5 (alternating series), 11.6, (Absolute conv, ratio and root), 11.7 (summary) in Calculus, Early Transcendentals (6th ed., 2008) by Stewart (Cengage) * sections 8.3 (Integral), 8.4 (Comparison), 8.5 (alternating), 8.6, Absolute conv, ratio and root, in Calculus, Early Transcendentals (1st ed., 2011) by Tan (Cengage) Integral tests, comparison tests, ratio & root tests.