Some Series and Integrals Involving the Riemann Zeta Function, Binomial Coefficients and the Harmonic Numbers
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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 . -
Conjugate Trigonometric Integrals! by R
SOME THEOREMS ON FOURIER TRANSFORMS AND CONJUGATE TRIGONOMETRIC INTEGRALS! BY R. P. BOAS, JR. 1. Introduction. Functions whose Fourier transforms vanish for large val- ues of the argument have been studied by Paley and Wiener.f It is natural, then, to ask what properties are possessed by functions whose Fourier trans- forms vanish over a finite interval. The two classes of functions are evidently closely related, since a function whose Fourier transform vanishes on ( —A, A) is the difference of two suitably chosen functions, one of which has its Fourier transform vanishing outside (—A, A). In this paper, however, we obtain, for a function having a Fourier transform vanishing over a finite interval, a criterion which does not seem to be trivially related to known results. It is stated in terms of a transform which is closely related to the Hilbert trans- form, and which may be written in the simplest (and probably most interest- ing) case, 1 C" /(* + *) - fix - 0 (1) /*(*) = —- sin t dt. J IT J o t2 With this notation, we shall show that a necessary and sufficient condition for a function/(x) belonging to L2(—<», oo) to have a Fourier transform vanishing almost everywhere on ( — 1, 1) is that/**(x) = —/(*) almost every- where;/**^) is the result of applying the transform (1) to/*(x). The result holds also if (sin t)/t is replaced in the integrand in (1) by a certain more general function Hit). As a preliminary result, we discuss the effect of the general transform/*(x) on a trigonometric Stieltjes integral, (2) /(*) = f e^'dait), J -R where a(/) is a normalized! complex-valued function of bounded variation. -
Topic 7 Notes 7 Taylor and Laurent Series
Topic 7 Notes Jeremy Orloff 7 Taylor and Laurent series 7.1 Introduction We originally defined an analytic function as one where the derivative, defined as a limit of ratios, existed. We went on to prove Cauchy's theorem and Cauchy's integral formula. These revealed some deep properties of analytic functions, e.g. the existence of derivatives of all orders. Our goal in this topic is to express analytic functions as infinite power series. This will lead us to Taylor series. When a complex function has an isolated singularity at a point we will replace Taylor series by Laurent series. Not surprisingly we will derive these series from Cauchy's integral formula. Although we come to power series representations after exploring other properties of analytic functions, they will be one of our main tools in understanding and computing with analytic functions. 7.2 Geometric series Having a detailed understanding of geometric series will enable us to use Cauchy's integral formula to understand power series representations of analytic functions. We start with the definition: Definition. A finite geometric series has one of the following (all equivalent) forms. 2 3 n Sn = a(1 + r + r + r + ::: + r ) = a + ar + ar2 + ar3 + ::: + arn n X = arj j=0 n X = a rj j=0 The number r is called the ratio of the geometric series because it is the ratio of consecutive terms of the series. Theorem. The sum of a finite geometric series is given by a(1 − rn+1) S = a(1 + r + r2 + r3 + ::: + rn) = : (1) n 1 − r Proof. -
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. -
Sequences and Series
From patterns to generalizations: sequences and series Concepts ■ Patterns You do not have to look far and wide to fi nd 1 ■ Generalization visual patterns—they are everywhere! Microconcepts ■ Arithmetic and geometric sequences ■ Arithmetic and geometric series ■ Common diff erence ■ Sigma notation ■ Common ratio ■ Sum of sequences ■ Binomial theorem ■ Proof ■ Sum to infi nity Can these patterns be explained mathematically? Can patterns be useful in real-life situations? What information would you require in order to choose the best loan off er? What other Draftscenarios could this be applied to? If you take out a loan to buy a car how can you determine the actual amount it will cost? 2 The diagrams shown here are the first four iterations of a fractal called the Koch snowflake. What do you notice about: • how each pattern is created from the previous one? • the perimeter as you move from the first iteration through the fourth iteration? How is it changing? • the area enclosed as you move from the first iteration to the fourth iteration? How is it changing? What changes would you expect in the fifth iteration? How would you measure the perimeter at the fifth iteration if the original triangle had sides of 1 m in length? If this process continues forever, how can an infinite perimeter enclose a finite area? Developing inquiry skills Does mathematics always reflect reality? Are fractals such as the Koch snowflake invented or discovered? Think about the questions in this opening problem and answer any you can. As you work through the chapter, you will gain mathematical knowledge and skills that will help you to answer them all. -
Mathematics 136 – Calculus 2 Summary of Trigonometric Substitutions October 17 and 18, 2016
Mathematics 136 – Calculus 2 Summary of Trigonometric Substitutions October 17 and 18, 2016 The trigonometric substitution method handles many integrals containing expressions like a2 x2, x2 + a2, x2 a2 p − p p − (possibly including expressions without the square roots!) The basis for this approach is the trigonometric identities 1 = sin2 θ + cos2 θ sec2 θ = tan2 θ +1. ⇒ from which we derive other related identities: a2 (a sin θ)2 = a2(1 sin2 θ) = √a2 cos2 θ = a cos θ p − q − (a tan θ)2 + a2 = a2(tan2 θ +1) = √a2 sec2 θ = a sec θ p q (a sec θ)2 a2 = a2(sec2 θ 1) = a2 tan2 θ = a tan θ p − p − p (Technical note: We usually assume that a > 0 and 0 <θ<π/2 here so that all the trig functions take positive values.) Hence, 1. If our integral contains √a2 x2, the substitution x = a sin θ will convert this radical to the simpler form a cos θ. − 2. If our integral contains √x2 + a2, the substitution x = a tan θ will convert this radical to the simpler form a sec θ. 3. If our integral contains √x2 a2, the substitution x = a sec θ will convert this radical to the simpler form a tan θ. − We substitute for the rest of the integral including the dx. For instance if x = a sin θ, then the dx = a cos θ dθ. If x = a tan θ, then dx = a sec2 θ dθ. If x = a sec θ, then dx = a sec θ tan θ dθ. -
Six Ways to Sum a Series Dan Kalman
Six Ways to Sum a Series Dan Kalman The College Mathematics Journal, November 1993, Volume 24, Number 5, pp. 402–421. Dan Kalman This fall I have joined the mathematics faculty at American University, Washington D. C. Prior to that I spent 8 years at the Aerospace Corporation in Los Angeles, where I worked on simulations of space systems and kept in touch with mathematics through the programs and publications of the MAA. At a national meeting I heard the presentation by Zagier referred to in the article. Convinced that this ingenious proof should be more widely known, I presented it at a meeting of the Southern California MAA section. Some enthusiastic members of the audience then shared their favorite proofs and references with me. These led to more articles and proofs, and brought me into contact with a realm of mathematics I never guessed existed. This paper is the result. he concept of an infinite sum is mysterious and intriguing. How can you add up an infinite number of terms? Yet, in some contexts, we are led to the Tcontemplation of an infinite sum quite naturally. For example, consider the calculation of a decimal expansion for 1y3. The long division algorithm generates an endlessly repeating sequence of steps, each of which adds one more 3 to the decimal expansion. We imagine the answer therefore to be an endless string of 3’s, which we write 0.333. .. In essence we are defining the decimal expansion of 1y3 as an infinite sum 1y3 5 0.3 1 0.03 1 0.003 1 0.0003 1 . -
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. -
Calculus Online Textbook Chapter 10
Contents CHAPTER 9 Polar Coordinates and Complex Numbers 9.1 Polar Coordinates 348 9.2 Polar Equations and Graphs 351 9.3 Slope, Length, and Area for Polar Curves 356 9.4 Complex Numbers 360 CHAPTER 10 Infinite Series 10.1 The Geometric Series 10.2 Convergence Tests: Positive Series 10.3 Convergence Tests: All Series 10.4 The Taylor Series for ex, sin x, and cos x 10.5 Power Series CHAPTER 11 Vectors and Matrices 11.1 Vectors and Dot Products 11.2 Planes and Projections 11.3 Cross Products and Determinants 11.4 Matrices and Linear Equations 11.5 Linear Algebra in Three Dimensions CHAPTER 12 Motion along a Curve 12.1 The Position Vector 446 12.2 Plane Motion: Projectiles and Cycloids 453 12.3 Tangent Vector and Normal Vector 459 12.4 Polar Coordinates and Planetary Motion 464 CHAPTER 13 Partial Derivatives 13.1 Surfaces and Level Curves 472 13.2 Partial Derivatives 475 13.3 Tangent Planes and Linear Approximations 480 13.4 Directional Derivatives and Gradients 490 13.5 The Chain Rule 497 13.6 Maxima, Minima, and Saddle Points 504 13.7 Constraints and Lagrange Multipliers 514 CHAPTER Infinite Series Infinite series can be a pleasure (sometimes). They throw a beautiful light on sin x and cos x. They give famous numbers like n and e. Usually they produce totally unknown functions-which might be good. But on the painful side is the fact that an infinite series has infinitely many terms. It is not easy to know the sum of those terms. -
Summation and Table of Finite Sums
SUMMATION A!D TABLE OF FI1ITE SUMS by ROBERT DELMER STALLE! A THESIS subnitted to OREGON STATE COLlEGE in partial fulfillment of the requirementh for the degree of MASTER OF ARTS June l94 APPROVED: Professor of Mathematics In Charge of Major Head of Deparent of Mathematics Chairman of School Graduate Committee Dean of the Graduate School ACKOEDGE!'T The writer dshes to eicpreßs his thanks to Dr. W. E. Mime, Head of the Department of Mathenatics, who has been a constant guide and inspiration in the writing of this thesis. TABLE OF CONTENTS I. i Finite calculus analogous to infinitesimal calculus. .. .. a .. .. e s 2 Suniming as the inverse of perfornungA............ 2 Theconstantofsuirrnation......................... 3 31nite calculus as a brancn of niathematics........ 4 Application of finite 5lflITh1tiOfl................... 5 II. LVELOPMENT OF SULTION FORiRLAS.................... 6 ttethods...........................a..........,.... 6 Three genera]. sum formulas........................ 6 III S1ThATION FORMULAS DERIVED B TIlE INVERSION OF A Z FELkTION....,..................,........... 7 s urnmation by parts..................15...... 7 Ratlona]. functions................................ Gamma and related functions........,........... 9 Ecponential and logarithrnic functions...... ... Thigonoretric arÎ hyperbolic functons..........,. J-3 Combinations of elementary functions......,..... 14 IV. SUMUATION BY IfTHODS OF APPDXIMATION..............,. 15 . a a Tewton s formula a a a S a C . a e a a s e a a a a . a a 15 Extensionofpartialsunmation................a... 15 Formulas relating a sum to an ifltegral..a.aaaaaaa. 16 Sumfromeverym'thterm........aa..a..aaa........ 17 V. TABLE OFST.Thß,..,,..,,...,.,,.....,....,,,........... 18 VI. SLThMTION OF A SPECIAL TYPE OF POER SERIES.......... 26 VI BIBLIOGRAPHY. a a a a a a a a a a . a . a a a I a s . -
Math 104 – Calculus 10.2 Infinite Series
Math 104 – Calculus 10.2 Infinite Series Math 104 - Yu InfiniteInfinite series series •• Given a sequence we try to make sense of the infinite Given a sequence we try to make sense of the infinite sum of its terms. sum of its terms. 1 • Example: a = n 2n 1 s = a = 1 1 2 1 1 s = a + a = + =0.75 2 1 2 2 4 1 1 1 s = a + a + a = + + =0.875 3 1 2 3 2 4 8 1 1 s = a + + a = + + =0.996 8 1 ··· 8 2 ··· 256 s20 =0.99999905 Nicolas Fraiman MathMath 104 - Yu 104 InfiniteInfinite series series Math 104 - YuNicolas Fraiman Math 104 Geometric series GeometricGeometric series series • A geometric series is one in which each term is obtained • A geometric series is one in which each term is obtained from the preceding one by multiplying it by the common ratiofrom rthe. preceding one by multiplying it by the common ratio r. 1 n 1 2 3 1 arn−1 = a + ar + ar2 + ar3 + ar − = a + ar + ar + ar + ··· kX=1 ··· kX=1 2 n 1 • DoesWe have not converges for= somea + ar values+ ar of+ r + ar − n ··· 1 2 3 n rsn =n ar1 + ar + ar + + ar r =1then ar − = a + a + a + a···+ n ···!1 sn rsn = a ar −kX=1 − 1 na(11 rn) r = 1 then s ar= − =− a + a a + a a − n 1 −r − − ··· kX=1 − Nicolas Fraiman MathNicolas 104 Fraiman Math 104 - YuMath 104 GeometricGeometric series series Nicolas Fraiman Math 104 - YuMath 104 Geometric series Geometric series 1 1 1 = 4n 3 n=1 X Math 104 - Yu Nicolas Fraiman Math 104 RepeatingRepeatingRepeang decimals decimals decimals • We cancan useuse geometricgeometric series series to to convert convert repeating repeating decimals toto fractions.fractions.