DOCSLIB.ORG

8. Uniform Convergence and Differentiation Suppose That (I) A

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
8. Uniform Convergence and Differentiation Suppose That (I) A 54 1. THE THEORY OF CONVERGENCE 8. Uniform convergence and differentiation Suppose that (i) a series ∞ u(x)= uk(x) Xk=1 converges pointwise on some interval [a,b]; 0 (ii) the derivatives un(x) are continuous on [a,b]; 0 (iii) the series of the derivatives k un converges uniformly on [a,b]. P Let us show that under these hypotheses the sum u(x) is continuously differentiable on [a,b] and ∞ ∞ d dun(x) (8.1) u0(x)= u (x)= dx n dx Xn=1 Xn=1 In other words, the order of differentiation and summation can be changed if the series of derivatives converges uniformly. Since the series of derivatives converges uniformly and the deriva- tives are continuous, then there exists a continuous function v(x) such that ∞ 0 v(x)= un(x) Xn=1 By the theorem about integration and uniform convergence, x ∞ x ∞ 0 v(t) dt = u (t) dt = un(x) un(a) Z Z n − a Xn=1 a Xn=1 = u(x) u(a) − for any x in [a,b], where the fundamental theorem of calculus and the convergence of the series un were used. Since v is continuous, its integral is differentiable withP respect to the upper limit and, hence, u is differentiable. Moreover, taking the derivative of both sides of the relation, x u(x)= u(a)+ v(t) dt Za it is concluded that u0(x)= v(x) which proves (8.1). It turns out that differentiability of u can also be established even 0 if the derivatives un(x) exist but are not necessarily continuous. In this 8. UNIFORM CONVERGENCE AND DIFFERENTIATION 55 case, the fundamental theorem of calculus cannot be used to establish differentiability of u(x). Nevertheless, the result still holds7: Theorem 8.1. Let un be a sequence of differentiable functions on { } [a,b]. Suppose that the sequence un(x0) converges for some x0 [a,b] and the sequence of derivatives {u0 (x) }converges uniformly on∈[a,b]. { n } Then the sequence un converges uniformly to some u(x) on [a,b], the limit function u(x){is differentiable} on [a,b] and 0 0 0 u (x)= lim un(x) = lim un(x) . n→∞ n→∞ Naturally, a similar result holds for series if the above theorem is applied to the sequence of partial sums. Example. The series ∞ sin2(nx) u(x)= 2+ n3 Xn=1 converges for all real x by the comparison test: 1 1 1 un(x) 3 3 and 3 < | |≤ 2+ n ≤ n X n ∞ In fact, it converges uniformly by the above argument. Each term of the series is a continuously differentiable function everywhere. The series of the derivatives converges uniformly everywhere because the series of upper bounds 0 n sin(2nx) n 1 1 un(x) = | 3 | 3 2 and 2 < | | 2+ n ≤ 2+ n ≤ n X n ∞ Therefore u(x) is continuously differentiable everywhere and ∞ ∞ n sin(2nx) u0(x)= u0 (x)= . n 2+ n3 Xn=1 Xn=1 8.1. Differentiability of a power series. The derivative of a function of a complex variable z is defined by f(z) f(w) f 0(z)= lim − w→z z w − In particular, (zn)0 = nzn−1 7see, e.g., W. Rudin, Principles of Mathematical Analysis, Chapter 7 56 1. THE THEORY OF CONVERGENCE because zn wn =(z w)(zn−1 + zn−2w + + wn−2z + wn−1) − − ··· Let f(z) be defined by a power series ∞ n f(z)= anz , z < R | | Xn=0 where R> 0 is the radius of convergence. The series of derivatives ∞ ∞ n−1 n annz = an+1(n + 1)z Xn=1 Xn=0 is also a power series with the same radius of convergence: n n n lim sup an+1 (n +1) = lim √n + 1 lim sup an+1 n→∞ n→∞ p| | n→∞ p| | n 1 = lim sup an = n→∞ R p| | Therefore it converges uniformly on any disk z b < R. Thus, f(z) is differentiable in the disk z < R and | |≤ | | ∞ 0 n−1 f (z)= annz , z < R | | Xn=1 The same argument can now be applied to the function g(z) = f 0(z) to conclude that ∞ 0 00 n−2 g (z)= f (z)= ann(n 1)z , z < R − | | Xn=2 In fact, it follows that a function defined by convergent power series is infinitely many times differentiable, its derivative are obtained term- by-term differentiation of the power series, ∞ (k) n−k f (z)= n(n 1) (n k)anz , z < R − ··· − | | Xn=k and these power series have the same radius of convergence. 8.2. Differentiation of a trigonometric Fourier series. Let f(x)bea2π periodic function whose derivatives are continuous (up to some order p).− By Fej´er’s theorem, its Fourier series converges to f(x) pointwise every- where: ∞ π ikx 1 −ikx f(x)= Ake , Ak = f(x)e dx 2π Z kX=−∞ −π 8. UNIFORM CONVERGENCE AND DIFFERENTIATION 57 Let us investigate how fast the Fourier coefficients Ak decay with in- creasing k . Using the integration by parts p times | | π π π 1 ikx 1 ikx 1 0 ikx Ak = f(x)e dx = f(x)e f (x)e dx 2π Z−π 2πik −π − 2πik Z−π π 1 = f 0(x)eikxdx −2πik Z−π 1 p 1 π = f (p)(x)eikxdx −ik 2π Z−π Since, by assumption, the derivative f (p)(x) is continuous and periodic, it is bounded on [ π,π] and it follows that − Mp (p) Ak p , Mp = sup f | |≤ k [−π,π] | | Thus, the Fourier coefficients of a function that has at least p continu- ous derivatives fall off at least as k−p with increasing k. In particular, if the function has infinitely many derivatives, then its Fourier coeffi- cients decay faster than any reciprocal power. This observation allows to conclude that the Fourier series of a function that is infinitely many times differentiable can be differenti- ated term-by-term any number of times and (p) p ikx f (x)= (ik) Ake Xk because the series converges uniformly for any p 0. Furthermore, suppose that f(x) is a real continuous,≥ 2π periodic function, and f 0(x) satisfies the hypotheses of Fej´er’s theorem− (see Sec- 0 tion 5.2). If Bk are the Fourier coefficients of f (x), then by integration by parts (as shown above) Bk Ak = Bk = ikAk , k = 0 − ik ⇒ − 6 π 0 Assuming that B0 = 0 (or −π f (x)dx = 0), it is concluded by Fej´er’s theorem that the series obtainedR by term-by-term differentiation of the Fourier series for f(x) converges pointwise: ikx ikx 1 0 0 ikAke = Bke = lim f (t)+ lim f (t) 2 t→x+ t→x− Xk Xk So, if the derivative f 0 is not continuous (and satisfies the above hy- potheses), the Fourier series obtained by term-by-term differentiation of the Fourier series for f converges to f 0 at each point where f 0 is continuous and to the mid-point of each jump discontinuity of f 0. 58 1. THE THEORY OF CONVERGENCE 8.3. Solutions to partial differential equations represented by Fourier series. Solutions to partial differential equations (PDEs) are often sought as series over some sets of functions (e.g., orthogonal sets of functions). In order to be a solution to a PDE, the sum of such a series should have continuous partial derivatives of sufficiently high order. For example, a solution to a wave equation 1 ∂2u ∂2u = 0 c2 ∂2t − ∂2x in two variables (x,t) should be at least twice continuously differen- tiable. If a solution is represented by a series of smooth functions, then differentiability of its sum is not obvious and requires an investigation. Suppose that periodic solutions to the above wave equation are sought: u(x + 2π,t)= u(x) The equation describes small vibrations of a circular string. Think of a circle of radius 1 as an equilibrium configuration of the string. The string can vibrate in the plane in which the circle lie and each point of the string can move only in the radial direction. In this case, u(x,t) is the shape of the vibrating string at a time moment t. By Fej´er’s theorem any continuous 2π periodic function can be represented by its trigonometric Fourier series− ∞ π ikt 1 −ikx u(x,t)= vk(t)e , vk(t)= u(x,t)e dx 2π Z kX=−∞ −π In particular, the sequence of its partial sums un(x,t) converges point- wise n ikt u(x,t)= lim un(x,t)= lim vk(t)e n→∞ n→∞ kX=−n for every real t. Let us find vk(t) for which a partial sum un(x,t) satisfies the wave equation. A physical solution is real. Owing to the linearity of the wave equation, a real solution can be obtained by taking either real or imaginary parts of a complex solution. A formulation using com- plex valued functions is technically simpler. Naturally, the periodicity ikx condition is fulfilled by un(x,t). Since the functions e are linearly independent, the coefficients vk(t) must satisfy the equation 1 v00(t)+ k2v (t) = 0 c2 k k 8. UNIFORM CONVERGENCE AND DIFFERENTIATION 59 Its general solution reads ikct −ikct vk(t)= ake + bke , k = 0 , v (t)= a + tb 6 0 0 0 where ak and bk are (complex) constants so that n ik(x+ct) ik(x−ct) un(x,t)= ake + bke kX=−n If ak and bk are such that the corresponding Fourier series converges, then it defines a function u(x,t).
Load more

Recommended publications
  • Ch. 15 Power Series, Taylor Series
    Ch. 15 Power Series, Taylor Series 서울대학교 조선해양공학과 서유택 2017.12 ※ 본 강의 자료는 이규열, 장범선, 노명일 교수님께서 만드신 자료를 바탕으로 일부 편집한 것입니다. Seoul National 1 Univ. 15.1 Sequences (수열), Series (급수), Convergence Tests (수렴판정) Sequences: Obtained by assigning to each positive integer n a number zn z . Term: zn z1, z 2, or z 1, z 2 , or briefly zn N . Real sequence (실수열): Sequence whose terms are real Convergence . Convergent sequence (수렴수열): Sequence that has a limit c limznn c or simply z c n . For every ε > 0, we can find N such that Convergent complex sequence |zn c | for all n N → all terms zn with n > N lie in the open disk of radius ε and center c. Divergent sequence (발산수열): Sequence that does not converge. Seoul National 2 Univ. 15.1 Sequences, Series, Convergence Tests Convergence . Convergent sequence: Sequence that has a limit c Ex. 1 Convergent and Divergent Sequences iin 11 Sequence i , , , , is convergent with limit 0. n 2 3 4 limznn c or simply z c n Sequence i n i , 1, i, 1, is divergent. n Sequence {zn} with zn = (1 + i ) is divergent. Seoul National 3 Univ. 15.1 Sequences, Series, Convergence Tests Theorem 1 Sequences of the Real and the Imaginary Parts . A sequence z1, z2, z3, … of complex numbers zn = xn + iyn converges to c = a + ib . if and only if the sequence of the real parts x1, x2, … converges to a . and the sequence of the imaginary parts y1, y2, … converges to b. Ex.
    [Show full text]
  • Formal Power Series - Wikipedia, the Free Encyclopedia
    Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series
    [Show full text]
  • 7. Properties of Uniformly Convergent Sequences
    46 1. THE THEORY OF CONVERGENCE 7. Properties of uniformly convergent sequences Here a relation between continuity, differentiability, and Riemann integrability of the sum of a functional series or the limit of a functional sequence and uniform convergence is studied. 7.1. Uniform convergence and continuity. Theorem 7.1. (Continuity of the sum of a series) N The sum of the series un(x) of terms continuous on D R is continuous if the series converges uniformly on D. ⊂ P Let Sn(x)= u1(x)+u2(x)+ +un(x) be a sequence of partial sum. It converges to some function S···(x) because every uniformly convergent series converges pointwise. Continuity of S at a point x means (by definition) lim S(y)= S(x) y→x Fix a number ε> 0. Then one can find a number δ such that S(x) S(y) <ε whenever 0 < x y <δ | − | | − | In other words, the values S(y) can get arbitrary close to S(x) and stay arbitrary close to it for all points y = x that are sufficiently close to x. Let us show that this condition follows6 from the hypotheses. Owing to the uniform convergence of the series, given ε > 0, one can find an integer m such that ε S(x) Sn(x) sup S Sn , x D, n m | − |≤ D | − |≤ 3 ∀ ∈ ∀ ≥ Note that m is independent of x. By continuity of Sn (as a finite sum of continuous functions), for n m, one can also find a number δ > 0 such that ≥ ε Sn(x) Sn(y) < whenever 0 < x y <δ | − | 3 | − | So, given ε> 0, the integer m is found.
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Uniform Convergence and Differentiation Theorem 6.3.1
    Math 341 Lecture #29 x6.3: Uniform Convergence and Differentiation We have seen that a pointwise converging sequence of continuous functions need not have a continuous limit function; we needed uniform convergence to get continuity of the limit function. What can we say about the differentiability of the limit function of a pointwise converging sequence of differentiable functions? 1+1=(2n−1) The sequence of differentiable hn(x) = x , x 2 [−1; 1], converges pointwise to the nondifferentiable h(x) = x; we will need to assume more about the pointwise converging sequence of differentiable functions to ensure that the limit function is differentiable. Theorem 6.3.1 (Differentiable Limit Theorem). Let fn ! f pointwise on the 0 closed interval [a; b], and assume that each fn is differentiable. If (fn) converges uniformly on [a; b] to a function g, then f is differentiable and f 0 = g. Proof. Let > 0 and fix c 2 [a; b]. Our goal is to show that f 0(c) exists and equals g(c). To this end, we will show the existence of δ > 0 such that for all 0 < jx − cj < δ, with x 2 [a; b], we have f(x) − f(c) − g(c) < x − c which implies that f(x) − f(c) f 0(c) = lim x!c x − c exists and is equal to g(c). The way forward is to replace (f(x) − f(c))=(x − c) − g(c) with expressions we can hopefully control: f(x) − f(c) f(x) − f(c) fn(x) − fn(c) fn(x) − fn(c) − g(c) = − + x − c x − c x − c x − c 0 0 − f (c) + f (c) − g(c) n n f(x) − f(c) fn(x) − fn(c) ≤ − x − c x − c fn(x) − fn(c) 0 0 + − f (c) + jf (c) − g(c)j: x − c n n The second and third expressions we can control respectively by the differentiability of 0 fn and the uniformly convergence of fn to g.
    [Show full text]
  • G: Uniform Convergence of Fourier Series
    G: Uniform Convergence of Fourier Series From previous work on the prototypical problem (and other problems) 8 < ut = Duxx 0 < x < l ; t > 0 u(0; t) = 0 = u(l; t) t > 0 (1) : u(x; 0) = f(x) 0 < x < l we developed a (formal) series solution 1 1 X X 2 2 2 nπx u(x; t) = u (x; t) = b e−n π Dt=l sin( ) ; (2) n n l n=1 n=1 2 R l nπy with bn = l 0 f(y) sin( l )dy. These are the Fourier sine coefficients for the initial data function f(x) on [0; l]. We have no real way to check that the series representation (2) is a solution to (1) because we do not know we can interchange differentiation and infinite summation. We have only assumed that up to now. In actuality, (2) makes sense as a solution to (1) if the series is uniformly convergent on [0; l] (and its derivatives also converges uniformly1). So we first discuss conditions for an infinite series to be differentiated (and integrated) term-by-term. This can be done if the infinite series and its derivatives converge uniformly. We list some results here that will establish this, but you should consult Appendix B on calculus facts, and review definitions of convergence of a series of numbers, absolute convergence of such a series, and uniform convergence of sequences and series of functions. Proofs of the following results can be found in any reasonable real analysis or advanced calculus textbook. 0.1 Differentiation and integration of infinite series Let I = [a; b] be any real interval.
    [Show full text]
  • Sequences, Series and Taylor Approximation (Ma2712b, MA2730)
    Sequences, Series and Taylor Approximation (MA2712b, MA2730) Level 2 Teaching Team Current curator: Simon Shaw November 20, 2015 Contents 0 Introduction, Overview 6 1 Taylor Polynomials 10 1.1 Lecture 1: Taylor Polynomials, Definition . .. 10 1.1.1 Reminder from Level 1 about Differentiable Functions . .. 11 1.1.2 Definition of Taylor Polynomials . 11 1.2 Lectures 2 and 3: Taylor Polynomials, Examples . ... 13 x 1.2.1 Example: Compute and plot Tnf for f(x) = e ............ 13 1.2.2 Example: Find the Maclaurin polynomials of f(x) = sin x ...... 14 2 1.2.3 Find the Maclaurin polynomial T11f for f(x) = sin(x ) ....... 15 1.2.4 QuestionsforChapter6: ErrorEstimates . 15 1.3 Lecture 4 and 5: Calculus of Taylor Polynomials . .. 17 1.3.1 GeneralResults............................... 17 1.4 Lecture 6: Various Applications of Taylor Polynomials . ... 22 1.4.1 RelativeExtrema .............................. 22 1.4.2 Limits .................................... 24 1.4.3 How to Calculate Complicated Taylor Polynomials? . 26 1.5 ExerciseSheet1................................... 29 1.5.1 ExerciseSheet1a .............................. 29 1.5.2 FeedbackforSheet1a ........................... 33 2 Real Sequences 40 2.1 Lecture 7: Definitions, Limit of a Sequence . ... 40 2.1.1 DefinitionofaSequence .......................... 40 2.1.2 LimitofaSequence............................. 41 2.1.3 Graphic Representations of Sequences . .. 43 2.2 Lecture 8: Algebra of Limits, Special Sequences . ..... 44 2.2.1 InfiniteLimits................................ 44 1 2.2.2 AlgebraofLimits.............................. 44 2.2.3 Some Standard Convergent Sequences . .. 46 2.3 Lecture 9: Bounded and Monotone Sequences . ..... 48 2.3.1 BoundedSequences............................. 48 2.3.2 Convergent Sequences and Closed Bounded Intervals . .... 48 2.4 Lecture10:MonotoneSequences .
    [Show full text]
  • Euler's Calculation of the Sum of the Reciprocals of the Squares
    Euler's Calculation of the Sum of the Reciprocals of the Squares Kenneth M. Monks ∗ August 5, 2019 A central theme of most second-semester calculus courses is that of infinite series. Simply put, to study infinite series is to attempt to answer the following question: What happens if you add up infinitely many numbers? How much sense humankind made of this question at different points throughout history depended enormously on what exactly those numbers being summed were. As far back as the third century bce, Greek mathematician and engineer Archimedes (287 bce{212 bce) used his method of exhaustion to carry out computations equivalent to the evaluation of an infinite geometric series in his work Quadrature of the Parabola [Archimedes, 1897]. Far more difficult than geometric series are p-series: series of the form 1 X 1 1 1 1 = 1 + + + + ··· np 2p 3p 4p n=1 for a real number p. Here we show the history of just two such series. In Section 1, we warm up with Nicole Oresme's treatment of the harmonic series, the p = 1 case.1 This will lessen the likelihood that we pull a muscle during our more intense Section 3 excursion: Euler's incredibly clever method for evaluating the p = 2 case. 1 Oresme and the Harmonic Series In roughly the year 1350 ce, a University of Paris scholar named Nicole Oresme2 (1323 ce{1382 ce) proved that the harmonic series does not sum to any finite value [Oresme, 1961]. Today, we would say the series diverges and that 1 1 1 1 + + + + ··· = 1: 2 3 4 His argument was brief and beautiful! Let us admire it below.
    [Show full text]
  • Series of Functions Theorem 6.4.2
    Math 341 Lecture #30 x6.4: Series of Functions Recall that we constructed the continuous nowhere differentiable function from Section 5.4 by using a series. We will develop the tools necessary to showing that this pointwise convergent series is indeed a continuous function. Definition 6.4.1. Let fn, n 2 N, and f be functions on A ⊆ R. The infinite series 1 X fn(x) = f1(x) + f2(x) + f3(x) + ··· n=1 converges pointwise on A to f(x) if the sequence of partial sums, sk(x) = f1(x) + f2(x) + ··· + fk(x) converges pointwise to f(x). The series converges uniformly on A to f if the sequence sk(x) converges uniformly on A to f(x). Uniform convergence of a series on A implies pointwise convergence of the series on A. For a pointwise or uniformly convergent series we write 1 1 X X f = fn or f(x) = fn(x): n=1 n=1 When the functions fn are continuous on A, each partial sum sk(x) is continuous on A by the Algebraic Continuity Theorem (Theorem 4.3.4). We can therefore apply the theory for uniformly convergent sequences to series. Theorem 6.4.2 (Term-by-term Continuity Theorem). Let fn be continuous functions on A ⊆ . If R 1 X fn n=1 converges uniformly to f on A, then f is continuous on A. Proof. We apply Theorem 6.2.6 to the partial sums sk = f1 + f2 + ··· + fk. When the functions fn are differentiable on a closed interval [a; b], we have that each partial sum sk is differentiable on [a; b] as well.
    [Show full text]
  • Classical Analysis
    Classical Analysis Edmund Y. M. Chiang December 3, 2009 Abstract This course assumes students to have mastered the knowledge of complex function theory in which the classical analysis is based. Either the reference book by Brown and Churchill [6] or Bak and Newman [4] can provide such a background knowledge. In the all-time classic \A Course of Modern Analysis" written by Whittaker and Watson [23] in 1902, the authors divded the content of their book into part I \The processes of analysis" and part II \The transcendental functions". The main theme of this course is to study some fundamentals of these classical transcendental functions which are used extensively in number theory, physics, engineering and other pure and applied areas. Contents 1 Separation of Variables of Helmholtz Equations 3 1.1 What is not known? . .8 2 Infinite Products 9 2.1 Definitions . .9 2.2 Cauchy criterion for infinite products . 10 2.3 Absolute and uniform convergence . 11 2.4 Associated series of logarithms . 14 2.5 Uniform convergence . 15 3 Gamma Function 20 3.1 Infinite product definition . 20 3.2 The Euler (Mascheroni) constant γ ............... 21 3.3 Weierstrass's definition . 22 3.4 A first order difference equation . 25 3.5 Integral representation definition . 25 3.6 Uniform convergence . 27 3.7 Analytic properties of Γ(z)................... 30 3.8 Tannery's theorem . 32 3.9 Integral representation . 33 3.10 The Eulerian integral of the first kind . 35 3.11 The Euler reflection formula . 37 3.12 Stirling's asymptotic formula . 40 3.13 Gauss's Multiplication Formula .
    [Show full text]