DOCSLIB.ORG

Summation of Trigonometric Series

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Summation of Trigonometric Series Dr. Dharmendra Kumar Yadav Assistant Professor of Mathematics M. L. S. College, Sarisab-Pahi, Madhubani R. K. College, Madhubani Lecture Notes Part-2 Paper-III B. Sc. Maths (H) CONVERGENCY AND DIVERGENCY OF INFINITE SERIES Part-1 Paper-I B. Sc. Maths (H) Summation of Trigonometric Series 01/04/2020 [M.L.S. College] 04/04/2020 [R. K. College] Part-2 Paper-III CONVERGENCY AND DIVERGENCY OF INFINITE SERIES Series: An expression in which the successive terms are formed by some regular law or definite law is called a series or simply the sum of the terms of a sequence is called a series. An expression of the form u1 + u2 + u3+…..un +……. is a series, in which u1, u2,..…un , ….are its terms. If the sum of the infinite series can be obtained it comes under convergent series otherwise under divergent or oscillatory series. Finite and Infinite Series: If a series has finite number of terms, it is called finite otherwise infinite. In finite series, the series terminates at some assigned term. In the infinite series, the series do not terminate. Note: Every finite series is a convergent series. Sum Notation: If {an} be the given series then an or an is denoted for infinite series. n1 Sum of Series: The sum of a finite series is finite because there is a last term in the case of a finite series. The sum of a finite series is the term by term addition. In the infinite series there is no last term so the process of addition will never be completed if we apply term by term addition. Hence in the ordinary case, the sum of an infinite series has no meaning. Therefore, the notion of limit has been employed to find out the sum of an infinite series. Convergent Series (Sum of Convergent Series): An infinite series is said to be convergent when the sum of the first n terms cannot numerically exceed some finite quantity however greater n may be, i.e. Sn ( Sn a1 a 2 a 3 .......... a n ) may tend to a definite finite number S such that when n is made sufficiently large, Sn-S is as small as we please. It may be written lim as SS , (where S a finite number), i.e. to every positive number 0 however n n small a positive integer m such that SSn when n m . lim Working Rule: If S exists and is equal to S a finite and unique quantity, then the n n series is called convergent and S is called its sum, otherwise the series is called divergent. If a series is convergent, its sum is finite. Divergent Series (Sum of Divergent Series): If the sum of an infinite series is greater than a finite quantity, i.e. if the sum is or , then the series is said to be divergent series. Generally, a positive term series which is not convergent is a divergent series. In other words, in an infinite series if lim lim S or S n n n n Then, the series is divergent. If a series is divergent, its sum is infinite. Example: 2n and n2 are divergent series. Oscillating Series (Sum of Oscillating Series): If the sum of an infinite series does not tend to a unique limit as n , finite or infinite, then the series is said to be an oscillating series. Example: ( 1)n1 . Note: If lim S n n is not unique, the series ∑an is known as oscillatory, sometimes called non-convergent series. Positive Term Series: An infinite series in which all the terms from and after some finite number of term are positive, is called a positive term infinite series or positive term series. 1 1 1 Example: 1 2 3 1 ............. 2 4 8 Fundamental Theorem of Infinite Series: The convergence or divergence of an infinite series remains unaffected by removal or addition of a finite number of terms. Note: Regrouping of terms does not affect the nature of the series. Part-1 Paper-I Summation of Trigonometric Series Generally we use two methods to find the summation of the trigonometric series: 1. Method of Differences, and 2. C + iS Method. Method of Differences We express each term of the series as the difference of two terms, so that one of the terms is the same function of the integer r as the other is of the (r-1) or (r+1). If we want to find the sum of the series u1 + u2 + u3 + … + un, We write ur = f(r+1) - f(r) or ur = f(r) - f(r-1). If we can write ur as ur = f(r+1) - f(r), then putting r = 1, 2, 3, … successively, we get u1 = f(2) - f(1) u2 = f(3) - f(2) u3 = f(4) - f(3) ……………… un-1 = f(n) - f(n-1) un = f(n+1) - f(n) Adding all we get u1 + u2 + u3 + … + un = f(n+1) - f(1) because the remaining terms cancel each other. Note: This method can also be used in finding out the sum of an infinite series, if the series is convergent. Thus if Sn be the sum to n terms of a given series, then the sum to infinity will be as follows lim lim S f (n 1) f (1). n n n Sum of sines of a Series of n Angles which are in Arithmetic Progression: n sin n 1 sin sin( ) sin( 2 ) ... sin{ (n 1)} 2 sin sin 2 2 Sum of cosines of a Series of n Angles which are in Arithmetic Progression: n sin n 1 cos cos( ) cos( 2 ) ... cos{ (n 1)} 2 cos 2 sin 2 C + iS Method: The sum of many of the trigonometric series can be found out by a general method which is called C + iS method. Let us suppose that we need to find out the sum of any one of the following series, whether finite or infinite: C a0 cos a1 cos( ) a2 cos( 2 ) ... and S a0 sin a1 sin( ) a2 sin( 2 ) ... The sine series is denoted by the first alphabet S and the cosine series is denoted by its first alphabet C. We multiply the sine series by i (iota) and then add it to the cosine series, we get C iS a0 cos a1 cos( ) a2 cos( 2 ) ... a0 sin a1 sin( ) a2 sin( 2 ) ... a0 (cos sin ) a1(cos( ) sin( )) a2 (cos( 2 ) sin( 2 )) ... i i( ) i( 2 ) a0e a1e a2e ... The last series is then transformed into one of the following series: 1. Arithmetic Series 2. Binomial Series 3. Exponential Series 4. Logarithmic Series 5. Sine or Cosine Series 6. Gregory Series Thus we express the sum of the given series in the form of A + iB and then equating the real and imaginary parts, we get the values of C and S. Previous Asked Questions in University Examinations 1. Find the sum of the sines of a series of angles which are in arithmetic progression. [LNMU 2019 (H)] 1 x 1 x 1 x 2. Sum the series tan x tan tan ... tan and find its limiting value when n 2 2 22 22 2n1 2n1 tends to infinity. [LNMU 2018(H)] 1 1 1 3. Sum the series cos cos(2 ) cos(3 ) cos(4 ) ...ad inf . [LNMU 2 3 4 2019(H)] .
Load more

Recommended publications
  • Blissard's Trigonometric Series with Closed-Form Sums
    BLISSARD’S TRIGONOMETRIC SERIES WITH CLOSED-FORM SUMS JACQUES GELINAS´ Abstract. This is a summary and verification of an elementary note written by John Blissard in 1862 for the Messenger of Mathematics. A general method of discovering trigonometric series having a closed-form sum is explained and illustrated with examples. We complete some state- ments and add references, using the summation symbol and Blissard’s own (umbral) representative notation for a more concise presentation than the original. More examples are also provided. 1. Historical examples Blissard’s well structured note [3] starts by recalling four trigonometric series which “mathemati- cal writers have exhibited as results of the differential and integral calculus” (A, B, C, G in the table below). Many such formulas had indeed been worked out by Daniel Bernoulli, Euler and Fourier. More examples can be found in 19th century textbooks and articles on calculus, in 20th century treatises on infinite series [21, 10, 19, 17] or on Fourier series [12, 23], and in mathematical tables [18, 22, 16, 11]. G.H. Hardy motivated the derivation of some simple formulas as follows [17, p. 2]. We can first agree that the sum of a geometric series 1+ x + x2 + ... with ratio x is s =1/(1 − x) because this is true when the series converges for |x| < 1, and “it would be very inconvenient if the formula varied in different cases”; moreover, “we should expect the sum s to satisfy the equation s = 1+ sx”. With x = eiθ, we obtain immediately a number of trigonometric series by separating the real and imaginary parts, by setting θ = 0, by differentiating, or by integrating [14, §13 ].
    [Show full text]
  • Power Series and Taylor's Series
    Power Series and Taylor’s Series Dr. Kamlesh Jangid Department of HEAS (Mathematics) Rajasthan Technical University, Kota-324010, India E-mail: [email protected] Dr. Kamlesh Jangid (RTU Kota) Series 1 / 19 Outline Outline 1 Introduction 2 Series 3 Convergence and Divergence of Series 4 Power Series 5 Taylor’s Series Dr. Kamlesh Jangid (RTU Kota) Series 2 / 19 Introduction Overview Everyone knows how to add two numbers together, or even several. But how do you add infinitely many numbers together ? In this lecture we answer this question, which is part of the theory of infinite sequences and series. An important application of this theory is a method for representing a known differentiable function f (x) as an infinite sum of powers of x, so it looks like a "polynomial with infinitely many terms." Dr. Kamlesh Jangid (RTU Kota) Series 3 / 19 Series Definition A sequence of real numbers is a function from the set N of natural numbers to the set R of real numbers. If f : N ! R is a sequence, and if an = f (n) for n 2 N, then we write the sequence f as fang. A sequence of real numbers is also called a real sequence. Example (i) fang with an = 1 for all n 2 N-a constant sequence. n−1 1 2 (ii) fang = f n g = f0; 2 ; 3 ; · · · g; n+1 1 1 1 (iii) fang = f(−1) n g = f1; − 2 ; 3 ; · · · g. Dr. Kamlesh Jangid (RTU Kota) Series 4 / 19 Series Definition A series of real numbers is an expression of the form a1 + a2 + a3 + ··· ; 1 X or more compactly as an, where fang is a sequence of real n=1 numbers.
    [Show full text]
  • 5. Sequences and Series of Functions in What Follows, It Is Assumed That X RN , and X a Means That the Euclidean Distance Between X and a Tends∈ to Zero,→X a 0
    32 1. THE THEORY OF CONVERGENCE 5. Sequences and series of functions In what follows, it is assumed that x RN , and x a means that the Euclidean distance between x and a tends∈ to zero,→x a 0. | − |→ 5.1. Pointwise convergence. Consider a sequence of functions un(x) (real or complex valued), n = 1, 2,.... The sequence un is said to converge pointwise to a function u on a set D if { } lim un(x)= u(x) , x D. n→∞ ∀ ∈ Similarly, the series un(x) is said to converge pointwise to a function u(x) on D RN if the sequence of partial sums converges pointwise to u(x): ⊂ P n Sn(x)= uk(x) , lim Sn(x)= u(x) , x D. n→∞ ∀ ∈ k=1 X 5.2. Trigonometric series. Consider a sequence an C where n = 0, 1, 2,.... The series { } ⊂ ± ± ∞ inx ane , x R ∈ n=−∞ X is called a complex trigonometric series. Its convergence means that sequences of partial sums m 0 + inx − inx Sm = ane , Sk = ane n=1 n=−k X X converge and, in this case, ∞ inx + − ane = lim Sm + lim Sk m→∞ k→∞ n=−∞ X ∞ ∞ Given two real sequences an and b , the series { }0 { }1 ∞ 1 a0 + an cos(nx)+ bn sin(nx) 2 n=1 X 1 is called a real trigonometric series. A factor 2 at a0 is a convention related to trigonometric Fourier series (see Remark below). For all x R for which a real (or complex) trigonometric series con- verges, the sum∈ defines a real-valued (or complex-valued) function of x.
    [Show full text]
  • IJR-1, Mathematics for All ... Syed Samsul Alam
    January 31, 2015 [IISRR-International Journal of Research ] MATHEMATICS FOR ALL AND FOREVER Prof. Syed Samsul Alam Former Vice-Chancellor Alaih University, Kolkata, India; Former Professor & Head, Department of Mathematics, IIT Kharagpur; Ch. Md Koya chair Professor, Mahatma Gandhi University, Kottayam, Kerala , Dr. S. N. Alam Assistant Professor, Department of Metallurgical and Materials Engineering, National Institute of Technology Rourkela, Rourkela, India This article briefly summarizes the journey of mathematics. The subject is expanding at a fast rate Abstract and it sometimes makes it essential to look back into the history of this marvelous subject. The pillars of this subject and their contributions have been briefly studied here. Since early civilization, mathematics has helped mankind solve very complicated problems. Mathematics has been a common language which has united mankind. Mathematics has been the heart of our education system right from the school level. Creating interest in this subject and making it friendlier to students’ right from early ages is essential. Understanding the subject as well as its history are both equally important. This article briefly discusses the ancient, the medieval, and the present age of mathematics and some notable mathematicians who belonged to these periods. Mathematics is the abstract study of different areas that include, but not limited to, numbers, 1.Introduction quantity, space, structure, and change. In other words, it is the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. Mathematicians seek out patterns and formulate new conjectures. They resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity.
    [Show full text]
  • Symmetric Integrals and Trigonometric Series
    SYMMETRIC INTEGRALS AND TRIGONOMETRIC SERIES George E. Cross and Brian S. Thomson 1 Introduction The first example of a convergent trigonometric series that cannot be ex- pressed in Fourier form is due to Fatou. The series ∞ sin nx (1) log(n + 1) nX=1 converges everywhere to a function that is not Lebesgue, or even Perron, integrable. It follows that the series (1) cannot be represented in Fourier form using the Lebesgue or Perron integrals. In fact this example is part of a whole class of examples as Denjoy [22, pp. 42–44] points out: if bn 0 ∞ ց and n=1 bn/n = + then the sum of the everywhere convergent series ∞ ∞ n=1Pbn sin nx is not Perron integrable. P The problem, suggested by these examples, of defining an integral so that the sum function f(x) of the convergent or summable trigonometric series ∞ (2) a0/2+ (an cos nx + bn sin nx) nX=1 is integrable and so that the coefficients, an and bn, can be written as Fourier coefficients of the function f has received considerable attention in the lit- erature (cf. [10], [11], [21], [22], [26], [32], [34], [37], [44] and [45].) For an excellent survey of the literature prior to 1955 see [27]; [28] is also useful. (In 1AMS 1980 Mathematics Subject Classification (1985 Revision). Primary 26A24, 26A39, 26A45; Secondary 42A24. 1 some of the earlier works ([10] and [45]) an additional condition on the series conjugate to (2) is imposed.) In addition a secondary literature has evolved devoted to the study of the properties of and the interrelations between the several integrals which have been constructed (for example [3], [4], [5], [6], [7], [8], [12], [13], [14], [15], [16], [17], [18], [19], [23], [29], [30], [31], [36], [38], [39], [40], [42] and [43]).
    [Show full text]
  • Trigonometric Series with General Monotone Coefficients
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector J. Math. Anal. Appl. 326 (2007) 721–735 www.elsevier.com/locate/jmaa Trigonometric series with general monotone coefficients ✩ S. Tikhonov Centre de Recerca Matemàtica (CRM), Bellaterra (Barcelona) E-08193, Spain Received 3 October 2005 Available online 18 April 2006 Submitted by D. Waterman Abstract We study trigonometric series with general monotone coefficients. Convergence results in the different metrics are obtained. Also, we prove a Hardy-type result on the behavior of the series near the origin. © 2006 Elsevier Inc. All rights reserved. Keywords: Trigonometric series; Convergence; Hardy–Littlewood theorem 1. Introduction In this paper we consider the series ∞ an cos nx (1) n=1 and ∞ an sin nx, (2) n=1 ✩ The paper was partially written while the author was staying at the Centre de Recerca Matemàtica in Barcelona, Spain. The research was funded by a EU Marie Curie fellowship (Contract MIF1-CT-2004-509465) and was partially supported by the Russian Foundation for Fundamental Research (Project 06-01-00268) and the Leading Scientific Schools (Grant NSH-4681.2006.1). E-mail address: [email protected]. 0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.02.053 722 S. Tikhonov / J. Math. Anal. Appl. 326 (2007) 721–735 { }∞ where an n=1 is given a null sequence of complex numbers. We define by f(x) and g(x) the sums of the series (1) and (2) respectively at the points where the series converge.
    [Show full text]
  • Chapter 1 Fourier Series
    Chapter 1 Fourier Series ßb 7W<ïtt|d³ ¯öóÍ ]y÷HÕ_ &tt d³Äó*× *y ÿ|1Òr÷bpÒr¸Þ ÷ßi;?÷ß×2 7W×ôT| T¹yd³JXà²ô :K» In this chapter we study Fourier series. Basic definitions and examples are given in Section 1. In Section 2 we prove the fundamental Riemann-Lebesgue lemma and discuss the Fourier series from the mapping point of view. Pointwise and uniform convergence of the Fourier series of a function to the function itself under various regularity assumptions are studied in Section 3. In Section 1.5 we establish the L2-convergence of the Fourier series without any additional regularity assumption. There are two applications. In Section 1.4 it is shown that every continuous function can be approximated by polynomials in a uniform manner. In Section 1.6 a proof of the classical isoperimetric problem for plane curves is presented. 1.1 Definition and Examples The concept of series of functions and their pointwise and uniform convergence were discussed in Mathematical Analysis II. Power series and trigonometric series are the most important classes of series of functions. We learned power series in Mathematical Analysis II and now we discuss Fourier series. First of all, a trigonometric series on [−π; π] is a series of functions of the form 1 X (an cos nx + bn sin nx); an; bn 2 R: n=0 1 2 CHAPTER 1. FOURIER SERIES As cos 0x = 1 and sin 0x = 0, we always set b0 = 0 and express the series as 1 X a0 + (an cos nx + bn sin nx): n=1 It is called a cosine series if all bn vanish and sine series if all an vanish.
    [Show full text]
  • Note on the Lerch Zeta Function
    Pacific Journal of Mathematics NOTE ON THE LERCH ZETA FUNCTION F. OBERHETTINGER Vol. 6, No. 1 November 1956 NOTE ON THE LERCH ZETA FUNCTION F. OBERHETTINGER l Introduction, The functional equation for the Riemann zeta function and Hurwitz's series were derived by a formal application of Poisson's summation formula in some previous papers [9], [10], [12, p. 17]. The results are correct although obtained by equating two infinite series whose regions of convergence exclude each other. It was shown later that this difficulty can be overcome if a generalized form of Poisson's formula is used [6]. It is the purpose of this note to show that the application of Poisson's summation formula in its ordinary form to the more general case of Lerch's zeta function [1], [2], [3], [5, pp. 27-30], [8] does not present the difficulties with respect to convergence which arise in its special cases. Thus a very simple and direct method for developing the theory of this function can be given. It may be noted that Hurwitz's series can be obtained immediately upon specializing one of the parameters, (cf. also [1]). 2. Lerch's zeta function Φ(z, s, v) is usually defined by the power series (la) Φ(z,8,v)= Σzm(v + mys \z\<l, vφO, -1, -2, . m=0 Equivalent to this is the integral representation [5, pp. 27-30] (lb) Φ(z, β, v)=— v~s + \Γ(v 2 Jo + 2 ("sin ( Jo \ \z\<\, which can be written as V]os (1c) Φ(z, 8, v)=—v- / iv- A- τΐ :-Γ(l-s)(logi) -^-V-5 Σ V; r^4- 4- 2 f °° sin f s tan-1-^— t log ^Vv2 4- t2)-sIZ(eZiίt - iyιdt Jo V v / upon replacing the first integral on the right of (lb) by the second and Received February 11, 1954.
    [Show full text]
  • INFINITE SERIES 1. Introduction the Two Basic Concepts of Calculus
    INFINITE SERIES KEITH CONRAD 1. Introduction The two basic concepts of calculus, differentiation and integration, are defined in terms of limits (Newton quotients and Riemann sums). In addition to these is a third fundamental limit process: infinite series. The label series is just another name for a sum. An infinite series is a \sum" with infinitely many terms, such as 1 1 1 1 (1.1) 1 + + + + ··· + + ··· : 4 9 16 n2 The idea of an infinite series is familiar from decimal expansions, for instance the expansion π = 3:14159265358979::: can be written as 1 4 1 5 9 2 6 5 3 5 8 π = 3 + + + + + + + + + + + + ··· ; 10 102 103 104 105 106 107 108 109 1010 1011 so π is an “infinite sum" of fractions. Decimal expansions like this show that an infinite series is not a paradoxical idea, although it may not be clear how to deal with non-decimal infinite series like (1.1) at the moment. Infinite series provide two conceptual insights into the nature of the basic functions met in high school (rational functions, trigonometric and inverse trigonometric functions, exponential and logarithmic functions). First of all, these functions can be expressed in terms of infinite series, and in this way all these functions can be approximated by polynomials, which are the simplest kinds of functions. That simpler functions can be used as approximations to more complicated functions lies behind the method which calculators and computers use to calculate approximate values of functions. The second insight we will have using infinite series is the close relationship between functions which seem at first to be quite different, such as exponential and trigonometric functions.
    [Show full text]
  • Summable Trigonometric Series
    Pacific Journal of Mathematics SUMMABLE TRIGONOMETRIC SERIES RALPH D. JAMES Vol. 6, No. 1 November 1956 SUMMABLE TRIGONOMETRIC SERIES R. D. JAMES l Introduction. One of the problems in the theory of trigono- metric series in the form (1.1) -—OD + Σ {an cos nx + bn sin nx) = Σ «n(aθ is that of suitably defining a process of integration such that, if the series (1.1) converges to a function f(x), then f(x) is integrable and the coefficients anf bn are given in Fourier form. The problem has been solved by Denjoy [3], Verblunsky [10], Marcinkiewicz and Zygmund [8], Burkill [1], [2], and James [6]. In Verblunsky's paper and in BurkilΓs first paper, additional hypotheses other than the convergence of (1.1) were made, and in all the papers some modification of the form of the Fourier formulas was necessary: An extension of the problem is to consider series that are summable (C, k), &2>1. This has been solved by Wolf [11] when the sum func- tion is Perron integrable. The problem of defining a process of inte- gration which may be applied to any series summable (C, k) may be solved if an additional condition involving the conjugate series a (1.2) Σ ( n sin nx - bn cos nx) = - is imposed. With this extra condition, it is proved, in § 2, that the formal product of cos px or sin px and a series summable (C, k) to f(x) is also summable (C, k) to f{x)c,o$px or /(a?) sin pa:.
    [Show full text]
  • On the Class of Limits of Lacunary Trigonometric Series
    On the class of limits of lacunary trigonometric series Christoph Aistleitner∗ Abstract Let (nk)k≥1 be a lacunary sequence of positive integers, i.e. a sequence satisfying 1 n +1/n >q> 1, k 1, and let f be a “nice” 1-periodic function with f(x) dx = 0. k k ≥ 0 Then the probabilistic behavior of the system (f(n x)) ≥1 is very similar to the behavior k k R of sequences of i.i.d. random variables. For example, Erd˝os and G´al proved in 1955 the following law of the iterated logarithm (LIL) for f(x) = cos2πx and lacunary (nk)k≥1: N −1/2 lim sup(2N log log N) f(nkx)= f 2 (1) N→∞ =1 k k Xk 1/2 1 2 for almost all x (0, 1), where f 2 = f(x) dx is the standard deviation of ∈ k k 0 the random variables f(nkx). If (nk)k≥1hasR certain number-theoretic properties (e.g. n +1/n ), a similar LIL holds for a large class of functions f, and the constant on k k → ∞ the right-hand side is always f 2. For general lacunary (nk)k≥1 this is not necessarily true: Erd˝os and Fortet constructedk k an example of a trigonometric polynomial f and a lacunary sequence (nk)k≥1, such that the lim sup in the LIL (1) is not equal to f 2 and not even a constant a.e. In this paper show that the class of possible functionsk k on the right-hand side of (1) can be very large: we give an example of a trigonometric polynomial f, such that for any function g(x) with sufficiently small Fourier coefficients there exists 2 a lacunary sequence (nk)k≥1 such that (1) holds with f 2 + g(x) instead of f 2 on the right-hand side.
    [Show full text]
  • 1 Taylor-Maclaurin Series
    1 Taylor-Maclaurin Series 2 00 2 Writing x = x0 + n4x; x1 = (n − 1)4x; ::, we get, (4y0) = y0 (4x) ::: and letting n ! 1, a leap of logic reduces the interpolation formula to: 1 y = y + (x − x )y0 + (x − x )2 y00 + ::: 0 0 0 0 2! 0 Definition 1.0.1. A function f is said to be an Cn function on (a; b) if f is n-times differentiable and the nth derivative f (n) is continuous on (a; b) and f is said to belong to C1 if every derivative of f exists (and is continuous) on (a; b). Taylor’s Formula: Suppose f belongs to C1 on (−R; R). Then for every n 2 N, and x 2 (−R; R) we have: 1 1 f(x) = f(0) + f 0(0)x + f (2)(0) x2 + ::: + f (n)(0) xn + R (x) 2! n! n f is said to be analytic at 0 if the remainder Rn(x) ! 0 as n ! 1. There are two standard forms for the remainder. 1. Integral form: Z x 1 (n+1) n Rn(x) = f (t)(x − t) dt n! 0 Proof. Integrating by parts, 1 Z x 1 1 Z x f (n+1)(t)(x − t)ndt = − f (n)(0)xn + f (n)(t)(x − t)n−1dt n! 0 n! (n − 1)! 0 Proof is completed by induction on n on observing that the result holds for n = 0 by the Funda- mental theorem of Calculus. 2. Lagrange’s form: There exists c 2 (0; x) such that: 1 R (x) = f (n+1)(c)xn+1 n (n + 1)! This form is easily derived from the integral form using intermediate value theorem of integral calculus.
    [Show full text]