Maclaurin Series

8 Key excerpts on "Maclaurin Series"

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  Mathematical Methods in Engineering and Physics

  • Gary N. Felder, Kenny M. Felder(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  Taylor series are the culmination of everything we have done so far in this chapter. ∙ A linear approximation is a special case of a Taylor series. If we take the linear approx- imation to the function 1∕(1 − x ) at x = 3, using the techniques developed earlier in the chapter, we get y = −1∕2 + (1∕4)(x − 3). With the Taylor series, however, we can add more terms to approximate the original function as accurately as we need. ∙ A Maclaurin Series is also a special case of a Taylor series. A Taylor series can be built to approximate a function around any specified x -value. If that value happens to be x = 0, then the Taylor series is a Maclaurin Series. Polynomials are among the easiest functions to work with. We can add and subtract them, compare them, take their derivatives, and integrate them in straightforward ways. The ability to replace any function with a polynomial is therefore a tremendously powerful mathemati- cal tool. Once again, however, a cautionary note must be sounded. Just because you can build a Taylor series does not mean that it works! The Taylor series we just built for 1∕(1 − x ) works wonderfully at x = 3.01;itapproximatesthefunctionverywellwithonlyafewterms.At x = 4 the series still works, but requires far more terms to create a reasonable approximation. And at x = 6 the series doesn’t work at all. (Try it and see what the terms are doing.) So there are important questions of domain that are taken up in the final sections of this chapter. A General Formula For Taylor Series In Problem 2.68, you will apply the Taylor series process described above to the generic func- tion f (x ). We can therefore write one concise formula 3 for the Taylor series of a function about x = a. f (x ) = ∞ ∑ n=0 f n (a) n! (x − a) n = f (a) + f ′ (a)(x − a) + f ′′ (a) 2 (x − a) 2 + f ′′′ (a) 6 (x − a) 3 … (2.4.1) Applying this formula directly takes you through the same steps we used above—the same derivatives and substitutions—in a way that you may find a bit quicker.

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  Calculus

  Single Variable

  • Carl V. Lutzer, H. T. Goodwill(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  Since f (k) (0) = 0 for all integer k ≥ 0, the Maclaurin Series is just 0 + 0t + 0t 2 + 0t 3 + · · · = 0. But f (t) = 0 at only the point t 0 = 0. You should understand Examples 8.14 and 8.15 as cautionary in nature, meant to indicate the need for further investigation and deeper understanding. As we’ll prove in just a moment, however, in the most important cases the value of a Taylor series agrees with the function used to generate it, enabling us to calculate val- ues of exponential and logarithmic functions, sine, and cosine using only the four arithmetic operations from elementary school. z Taylor’s Theorem A moment ago we said that the Taylor series generated by f is equal to f (t) in the most important cases, and now we prove that assertion. The first step in our proof will be using integration by parts to develop the formula f (t) = f (0) + f 0 (0)t + f 00 (0) 2! t 2 + · · · + f (n) (0) n! t n + (-1) n Z t 0 (x - t) n n! f (n+1) (x) dx, in which you see f (t) expressed as the sum of a definite integral and the n th -degree Taylor polynomial. As n → ∞ the Taylor polynomial becomes the Taylor series, so the conclusion that f (t) equals its Taylor series will rest on whether the integral term vanishes in the limit.  Integrating by parts to generate a series We know from the Fundamental Theorem of Calculus that f (t) = f (0) + Z t 0 f 0 (x) dx (8.3) provided that f 0 is continuous over [0, t], which we’ll assume for the sake of discus- sion. A moment ago we indicated that we would integrate by parts, which we do Section 8.8 Taylor and Maclaurin Series 676 by writing the integrand as 1 · f 0 (x). Then we set u = f 0 (x) and v 0 = 1. This would typically lead us to write v = x, which is the difference between x and the lower bound of integration in this case (v = x - 0); but v 0 = 1 also allows us to define v as the difference between x and the upper bound of integration, v = x - t, and this turns out to be a better choice.

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  Vector Calculus

  • William Cox(Author)
  • 1998(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  6.1 Introduction (i) You probably gave the trigonometric definition of sin x sin x = Opposite Hypotenuse and the series definition (ii) For eX you might have given the limit definition eX = lim (1 +~) n n--:,oo n and again you probably gave the series definition X x 2 x 3 e = 1 +x+2f+3f+ ... These examples are enough to show the utility and power of the series definition for functions of a single variable. Such definitions are useful from the theoretical point of view because they provide a uniform means of expressing any function in terms of sums (albeit infinite sums) of powers. They are important from a practical point of view because they provide a means of evaluating a function to any required accuracy. The key point about such a series (technically, a Maclaurin Series -or Taylor series about the origin x = 0) is that the only terms appearing are powers of x. In general, f(x) = a o + a1x + a 2x 2 + ... + a.x' + ... Write down the typical term you might expect in a series expansion of a function of two variablesf(x, y). Taylor Series for Functions of Several Variables 77 It does not take much imagination to guess that you will get terms like xrys for all integer values of rand s. This would be for 'expansion about the origin'. A general series might start something like lex, y) = aOO + alOx + aOlY + a 20x 2 + allxy + ao2y2 + ... The pattern is fairly obvious -but how would we find the coefficients aij? Remember that in the case of a function of a single variable the coefficients are given by differentiation for example (see Pearson, 1996), So you can bet that the aij are likewise going to involve derivatives, but partial derivatives. You may also remember that in the rigorous formal study of Taylor and Maclaurin Series of a single variable we avoided the sloppiness of the' + ... ' at the end of the series by using a remainder term which tended to zero as we took more and more terms of the series (see Kopp, 1996).

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  Multivariable Calculus, Metric Edition

  • James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  As we saw in the examples of Section 11.9, we can replace x in a given Taylor series by an expression of the form cx m , we can multiply (or divide) the series by such an expression, and we can differentiate or integrate term by term (Theorem 11.9.2). It can be shown that we can also obtain new Taylor series by adding, subtracting, multiplying, or dividing Taylor series. EXAMPLE 10 Find the Maclaurin Series for (a) f s xd - x cos x and (b) f s xd - lns1 1 3x 2 d. SOLUTION (a) We multiply the Maclaurin Series for cos x (see Table 1) by x : x cos x - x o ` n-0 s21d n x 2 n s2nd! - o ` n-0 s21d n x 2 n11 s2nd! for all x (b) Replacing x by 3x 2 in the Maclaurin Series for lns1 1 xd gives lns1 1 3x 2 d - o ` n-1 s21d n21 s3x 2 d n n - o ` n-1 s21d n21 3 n x 2 n n We know from Table 1 that this series converges for | 3x 2 | , 1, that is | x | , 1ys3 , so the radius of convergence is R - 1ys3 . ■ EXAMPLE 11 Find the function represented by the power series o ` n-0 s21d n 2 n x n n! . SOLUTION By writing o ` n-0 s21d n 2 n x n n! - o ` n-0 s22xd n n! we see that this series is obtained by replacing x with 22x in the series for e x (in Table 1). Thus the series represents the function e 22x . ■ EXAMPLE 12 Find the sum of the series 1 1  2 2 1 2  2 2 1 1 3  2 3 2 1 4  2 4 1 ∙ ∙ ∙. SOLUTION With sigma notation we can write the given series as o ` n-1 s21d n21 1 n  2 n - o ` n-1 s21d n21 ( 1 2 ) n n Then from Table 1 we see that this series matches the entry for lns1 1 xd with x - 1 2 . So o ` n-1 s21d n21 1 n  2 n - ln(1 1 1 2 ) - ln 3 2  ■ One reason that Taylor series are important is that they enable us to integrate functions that we couldn’t previously handle. In fact, in the introduction to this chapter we men- tioned that Newton often integrated functions by first expressing them as power series and Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

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  Calculus of a Single Variable: Early Transcendental Functions, International Metric Edition

  • Ron Larson, Bruce Edwards(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  For instance, the function f (x) = e x can be approximated by its third-degree Maclaurin polynomial e x ≈ 1 + x + x 2 2! + x 3 3! . In that section, you saw that the higher the degree of the approximating polynomial, the better the approximation becomes. In this and the next two sections, you will see that several important types of functions, including f (x) = e x , can be represented exactly by an infinite series called a power series. For example, the power series representation for e x is e x = 1 + x + x 2 2! + x 3 3! + . . . + x n n! + . . . . For each real number x, it can be shown that the infinite series on the right converges to the number e x . Before doing this, however, some preliminary results dealing with power series will be discussed—beginning with the next definition. Definition of Power Series If x is a variable, then an infinite series of the form ∑ ∞ n =0 a n x n = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + . . . + a n x n + . . . is called a power series. More generally, an infinite series of the form ∑ ∞ n =0 a n (x - c) n = a 0 + a 1 (x - c) + a 2 (x - c) 2 + . . . + a n (x - c) n + . . . is called a power series centered at c, where c is a constant. Power Series a. The following power series is centered at 0. ∑ ∞ n =0 x n n! = 1 + x + x 2 2 + x 3 3! + . . . b. The following power series is centered at -1. ∑ ∞ n =0 (-1) n (x + 1) n = 1 - (x + 1) + (x + 1) 2 - (x + 1) 3 + . . . c. The following power series is centered at 1. ∑ ∞ n =1 1 n (x - 1) n = (x - 1) + 1 2 (x - 1) 2 + 1 3 (x - 1) 3 + . . . REMARK To simplify the notation for power series, assume that (x - c) 0 = 1, even when x = c. Exploration Graphical Reasoning Use a graphing utility to approximate the graph of each power series near x = 0. (Use the first several terms of each series.) Each series represents a well-known function. What is the function? a. ∑ ∞ n =0 (-1) n x n n! b. ∑ ∞ n =0 (-1) n x 2n (2n)! c. ∑ ∞ n =0 (-1) n x 2n +1 (2n + 1)! d. ∑ ∞ n =0 (-1) n x 2n +1 2n + 1 e.

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  Foundation Mathematics for the Physical Sciences

  • K. F. Riley, M. P. Hobson(Authors)
  • 2011(Publication Date)
  • Cambridge University Press

    (Publisher)

  It is in fact the Maclaurin expansion of exp x (see Section 6.6.3 and Appendix A ). Therefore, S ( x ) = exp x and, of course, its value varies according to the value of the variable x . A series might just as easily depend on a complex variable z . A general, random sequence of numbers can be described as a series and a sum of the terms found. However, for cases of practical interest, there will usually be some sort of pattern in the form of the u n – typically that u n is a function of n – and hence a relationship between successive terms. 1 For example, if the n th term of a series is given by u n = 1 2 n , • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 1 Write u n as a function of n for the series S ( x ) = exp x and state the relationship between successive terms. Do the same for S ( x ) = e − x . 213 214 Series and limits for n = 1 , 2 , 3 , . . . , N , then the sum of the first N terms will be S N = N n = 1 u n = 1 2 + 1 4 + 1 8 + · · · + 1 2 N . (6.1) It is clear that the sum of a finite number of terms is always finite, provided that each term is itself finite. It is often of practical interest, however, to consider the sum of a series with an infinite number of finite terms. The sum of an infinite number of terms is best defined by first considering the partial sum of the first N terms, S N . If the value of the partial sum S N tends to a finite limit, S , as N tends to infinity, then the series is said to converge and its sum is given by the limit S . In other words, the sum of an infinite series is given by S = lim N →∞ S N , provided the limit exists. 2 For complex infinite series, if S N approaches a limit S = X + iY as N → ∞ , this means that X N → X and Y N → Y separately, i.e. the real and imaginary parts of the series are each convergent series with sums X and Y respectively.

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  Sixth Form Pure Mathematics

  Volume 1

  • C. Plumpton, W. A. Tomkys(Authors)
  • 2014(Publication Date)
  • Pergamon

    (Publisher)

  In each case, the con-ditions of convergence will be merely stated. In a subsequent volume we shall discuss in detail the conditions of convergence and the assumptions which are made here concerning the conditions in particular cases will be justified. 9.2 Maclaurin's Expansion We assume that, if f{x) can be expanded as an infinite, convergent, series of powers of x y then m . /(o )+ r(o)* + q^ + .. . + ™*+ .... (9.2, This is known as Maclaurin's Expansion. For most purposes, the basic assumptions concerning convergence made above will not be involved in the use made of series expansions since we are concerned only with the first few terms of the expansion as an approximation to the function and not with the expansion as an infinite series which is equivalent to the function. Example. Use the Maclaurin expansion to obtain an approximate value of sin x in terms of x when x is so small that terms of higher power BINOMIAL, EXPONENTIAL AND LOGARITHM FUNCTIONS 331 than JC 3 may be neglected. If /(*) = sinx, /(0) = 0, f(x) = cos x, f(0)=l 9 r w = -smx, r(o) = o, /'(*) = -cosx, / , , , (0)= -1 . x z Therefore, for small values of x, sin x 4= x— jj • The infinite series for sin x and cos x are x 3 x 5 x 7 (-l) n x 2n + 1 / n „ x M n 3 C a s X 3 ! + S!7l + --+ (2.4-1)! + — ( 9 3 ) Y 2 v4 v * /'—l^nv^ c o s , = i _ | . + | ! _ | . + . . . + l _ ^ _ + .... (9.4) These expansions are valid for all values of x. Exercises 9.2 Use the Maclaurin Expansion to obtain approximate expansions for small values of x as far as the term in JC 8 in each of the following cases. 1. (I-*) 1 /*. 2. (I-* 2 )-1 /*. 3. cos*. 4. tan*. 9.3 The Binomial Series If /(*) = (! + *)*, then f(x) = n(l+xr-i, /(*) =/K-i)(i+*) n -2 , /'(*) = / K / I -1 ) ( H -2 ) ( 1 + X)»-3 . f(jc) = « ( » -l ) ( » -2 ) . . . M l ) ( l + ^ . 332 Hence SIXTH FORM PURE MA' /(0) = 1, /'(0) = n, /(0) = «(«-l), /'(O) = «(-!)(»-2), y>(0) -n(n-)(n-2) .

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  Calculus

  Single Variable

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  Thus, we can be certain that the series we have obtained converges for all x in the interval −1 ≤ x ≤ 1 (why?). 2x 5 15 2x 5 15 x 3 3 x + + + . . . x 5 120 x 3 6 x – + – . . . x 5 24 x 3 2 x – + – . . . x 5 30 x 3 3 – + . . . x 5 6 x 3 3 – + . . . + . . . x 4 24 x 2 2 1 – + – . . . Example 6 Find the first three nonzero terms in the Maclaurin Series for tan x. Solution Using the first three terms in the Maclaurin Series for sin x and cos x, we can express tan x as tan x = sin x cos x = x − x 3 3! + x 5 5! − · · · 1 − x 2 2! + x 4 4! − · · · Dividing, as shown in the margin, we obtain tan x = x + x 3 3 + 2x 5 15 + · · · Modeling Physical Laws with Taylor Series Taylor series provide an important way of modeling physical laws. To illustrate the idea we will consider the problem of modeling the period of a simple pendulum (Figure 9.10.1). As explained in Chapter 7 Making Connections Exercise 5, the period T of such a pendulum is given by T = 4  L g  π/2 0 1  1 − k 2 sin 2 φ dφ (7) L θ 0 FIGURE 9.10.1 where L = length of the supporting rod g = acceleration due to gravity k = sin (θ 0 / 2), where θ 0 is the initial angle of displacement from the vertical The integral, which is called a complete elliptic integral of the first kind, cannot be expressed in terms of elementary functions and is often approximated by numerical methods. Unfortunately, numerical values are so specific that they often give little insight into general physical principles. However, if we expand the integrand of (7) in a series and integrate term by term, then we can generate an infinite series that can be used to construct various mathematical models for the period T that give a deeper understanding of the behavior of the pendulum.

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