2. Infinite Series
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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 = . -
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]. -
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. -
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 →∞. -
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. -
Almost Periodic Sequences and Functions with Given Values
ARCHIVUM MATHEMATICUM (BRNO) Tomus 47 (2011), 1–16 ALMOST PERIODIC SEQUENCES AND FUNCTIONS WITH GIVEN VALUES Michal Veselý Abstract. We present a method for constructing almost periodic sequences and functions with values in a metric space. Applying this method, we find almost periodic sequences and functions with prescribed values. Especially, for any totally bounded countable set X in a metric space, it is proved the existence of an almost periodic sequence {ψk}k∈Z such that {ψk; k ∈ Z} = X and ψk = ψk+lq(k), l ∈ Z for all k and some q(k) ∈ N which depends on k. 1. Introduction The aim of this paper is to construct almost periodic sequences and functions which attain values in a metric space. More precisely, our aim is to find almost periodic sequences and functions whose ranges contain or consist of arbitrarily given subsets of the metric space satisfying certain conditions. We are motivated by the paper [3] where a similar problem is investigated for real-valued sequences. In that paper, using an explicit construction, it is shown that, for any bounded countable set of real numbers, there exists an almost periodic sequence whose range is this set and which attains each value in this set periodically. We will extend this result to sequences attaining values in any metric space. Concerning almost periodic sequences with indices k ∈ N (or asymptotically almost periodic sequences), we refer to [4] where it is proved that, for any precompact sequence {xk}k∈N in a metric space X , there exists a permutation P of the set of positive integers such that the sequence {xP (k)}k∈N is almost periodic. -
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. -
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. -
Discrete Signals and Their Frequency Analysis
Discrete signals and their frequency analysis. Valentina Hubeika, Jan Cernock´yˇ DCGM FIT BUT Brno, {ihubeika,cernocky}@fit.vutbr.cz • recapitulation – fundamentals on discrete signals. • periodic and harmonic sequences • discrete signal processing • convolution • Fourier transform with discrete time • Discrete Fourier Transform 1 Sampled signal ⇒ discrete signal During sampling we consider only values of the signal at sampling period multiplies T : x(nT ), nT = ... − 2T, −T, 0,T, 2T, 3T,... For a discrete signal, we forget about real time and simply count the samples. Discrete time becomes : x[n], n = ... − 2, −1, 0, 1, 2, 3,... Thus we often call discrete signals just sequences. 2 Important discrete signals Unit step and unit impulse: 1 for n ≥ 0 1 for n =0 σ[n]= δ[n]= 0 elsewhere 0 elsewhere 3 Periodic discrete signals their behaviour repeats after N samples, the smallest possible N is denoted as N1 and is called fundamental period. Harmonic discrete signals (harmonic sequences) x[n]= C1 cos(ω1n + φ1) (1) • C1 is a positive constant – magnitude. • ω1 is a spositive constant – normalized angular frequency. As n is just a number, the unit of ω1 is [rad]. Note, that in the previous lecture we denoted with the same simbol an angular frequency of continuous signals. Although in the last lecture we used ′ symbol ω1 for discrete time, we will not do it any longer. You will recognize continuous time frequency if there is real time associated with it (for instance cos(ω1t)). If you see discrete time n (for instance cos(ω1n)) you should know we are talking about normalized angular frequency. -
Arxiv:1904.09040V2 [Math.NT]
PERIODICITIES FOR TAYLOR COEFFICIENTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS PAVEL GUERZHOY, MICHAEL H. MERTENS, AND LARRY ROLEN Abstract. Congruences of Fourier coefficients of modular forms have long been an object of central study. By comparison, the arithmetic of other expansions of modular forms, in particular Taylor expansions around points in the upper-half plane, has been much less studied. Recently, Romik made a conjecture about the periodicity of coefficients around τ0 = i of the classical Jacobi theta function θ3. Here, we generalize the phenomenon observed by Romik to a broader class of modular forms of half-integral weight and, in particular, prove the conjecture. 1. Introduction Fourier coefficients of modular forms are well-known to encode many interesting quantities, such as the number of points on elliptic curves over finite fields, partition numbers, divisor sums, and many more. Thanks to these connections, the arithmetic of modular form Fourier coefficients has long enjoyed a broad study, and remains a very active field today. However, Fourier expansions are just one sort of canonical expansion of modular forms. Petersson also defined [17] the so-called hyperbolic and elliptic expansions, which instead of being associated to a cusp of the modular curve, are associated to a pair of real quadratic numbers or a point in the upper half-plane, respectively. A beautiful exposition on these different expansions and some of their more recent connections can be found in [8]. In particular, there Imamoglu and O’Sullivan point out that Poincar´eseries with respect to hyperbolic expansions include the important examples of Katok [11] and Zagier [26], which are the functions which Kohnen later used [15] to construct the holomorphic kernel for the Shimura/Shintani lift. -
Construction of Almost Periodic Sequences with Given Properties
Electronic Journal of Differential Equations, Vol. 2008(2008), No. 126, pp. 1–22. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) CONSTRUCTION OF ALMOST PERIODIC SEQUENCES WITH GIVEN PROPERTIES MICHAL VESELY´ Abstract. We define almost periodic sequences with values in a pseudometric space X and we modify the Bochner definition of almost periodicity so that it remains equivalent with the Bohr definition. Then, we present one (easily modifiable) method for constructing almost periodic sequences in X . Using this method, we find almost periodic homogeneous linear difference systems that do not have any non-trivial almost periodic solution. We treat this problem in a general setting where we suppose that entries of matrices in linear systems belong to a ring with a unit. 1. Introduction First of all we mention the article [9] by Fan which considers almost periodic sequences of elements of a metric space and the article [22] by Tornehave about almost periodic functions of the real variable with values in a metric space. In these papers, it is shown that many theorems that are valid for complex valued sequences and functions are no longer true. For continuous functions, it was observed that the important property is the local connection by arcs of the space of values and also its completeness. However, we will not use their results or other theorems and we will define the notion of the almost periodicity of sequences in pseudometric spaces without any conditions, i.e., the definition is similar to the classical definition of Bohr, the modulus being replaced by the distance. -
Florentin Smarandache
FLORENTIN SMARANDACHE SEQUENCES OF NUMBERS INVOLVED IN UNSOLVED PROBLEMS 1 141 1 1 1 8 1 1 8 40 8 1 1 8 1 1 1 1 12 1 1 12 108 12 1 1 12 108 540 108 12 1 1 12 108 12 1 1 12 1 1 1 1 16 1 1 16 208 16 1 1 16 208 1872 208 16 1 1 16 208 1872 9360 1872 208 16 1 1 16 208 1872 208 16 1 1 16 208 16 1 1 16 1 1 2006 Introduction Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. These notions, definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc. ( on integer sequences, numbers, quotients, residues, exponents, sieves, pseudo-primes/squares/cubes/factorials, almost primes, mobile periodicals, functions, tables, prime/square/factorial bases, generalized factorials, generalized palindromes, etc. ) have been extracted from the Archives of American Mathematics (University of Texas at Austin) and Arizona State University (Tempe): "The Florentin Smarandache papers" special collections, University of Craiova Library, and Arhivele Statului (Filiala Vâlcea, Romania). It is based on the old article “Properties of Numbers” (1975), updated many times. Special thanks to C. Dumitrescu & V. Seleacu from the University of Craiova (see their edited book "Some Notions and Questions in Number Theory", Erhus Univ. Press, Glendale, 1994), M. Perez, J. Castillo, M. Bencze, L. Tutescu, E, Burton who helped in collecting and editing this material. The Author 1 Sequences of Numbers Involved in Unsolved Problems Here it is a long list of sequences, functions, unsolved problems, conjectures, theorems, relationships, operations, etc.