Taylor Series

7 Key excerpts on "Taylor Series"

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  Calculus

  The Elements

  • Michael Comenetz(Author)
  • 2002(Publication Date)
  • WSPC

    (Publisher)

  CHAPTER 8 Representation of Functions by Infinite Taylor Series INTRODUCTION The polynomials are a large and varied class of functions formed by the simplest arithmetic operations and presenting no obstacle to the performance of differentiation and integration. Although a natural wish that every func-tion might be a polynomial goes unrealized, we will see in this chapter that most ordinary functions can be arbitrarily well approximated by polynomials of successively higher degrees, and this in a systematic way. There results a uniform representation of functions as polynomial-like expressions of infinitely many terms, somewhat analogous to the decimal system by which numbers are expressed as infinite “sums” of powers of ten with coefficients. The benefits for both understanding and calculation are very great. In the first article this representation by the Taylor Series, as it is called, is derived from polynomial approximations, and evidence of its value is provided by an uncritical look at the series associated with the exponential function and the sine and cosine. The second article begins with a theorem by which the Taylor Series can be justified, and goes on to establish the validity of several important series by means of the theorem and certain special arguments. 8.1 POLYNOMIAL APPROXIMATION AND THE Taylor Series 8.1.1 The first three approximating polynomials. A. Constant and linear approximation. I will begin with a review of the approxima-tions considered in §§ 3.7.11 and 3.8.6–8, with somewhat different emphasis. 1 1 It is advisable to read or re-read those sections. 389 390 REPRESENTATION BY INFINITE SERIES § 8.1.1 Suppose that the value of a continuous function f ( x ) is known at x = a , but unknown elsewhere. Because of continuity, it is reasonable to estimate its value at points x near a by f ( x ) ≈ f ( a ) .

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  Calculus

  Single Variable

  • Deborah Hughes-Hallett, William G. McCallum, Andrew M. Gleason, Eric Connally, Daniel E. Flath, Selin Kalaycioglu, Brigitte Lahme, Patti Frazer Lock, David O. Lomen, David Lovelock, Guadalupe I. Lozano, Jerry Morris, Brad G. Osgood, Cody L. Patterson, Douglas Quinney, Karen R. Rhea, Ayse Arzu Sahin, Adam H. Spiegler(Authors)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  Assuming the function has all its derivatives defined, finding the coefficients can always be done, in theory at least, by differentiation. That is how we derived the four most important Taylor Series, those for the functions   , sin , cos , and (1 + )  . For many functions, however, computing Taylor Series coefficients by differentiation can be a very laborious business. We now introduce easier ways of finding Taylor Series, if the series we want is closely related to a series that we already know. New Series by Substitution Suppose we want to find the Taylor Series for  − 2 about  = 0. We could find the coefficients by differentiation. Differentiating  − 2 by the chain rule gives −2 − 2 , and differentiating again gives −2 − 2 + 4 2  − 2 . Each time we differentiate we use the product rule, and the number of terms grows. Finding the tenth or twentieth derivative of  − 2 , and thus the series for  − 2 up to the  10 or  20 terms, by this method is tiresome (at least without a computer or calculator that can differentiate). Fortunately, there’s a quicker way. Recall that   = 1 +  +  2 2! +  3 3! +  4 4! + ⋯ for all . Substituting  = − 2 tells us that  − 2 = 1 + (− 2 ) + (− 2 ) 2 2! + (− 2 ) 3 3! + (− 2 ) 4 4! + ⋯ = 1 −  2 +  4 2! −  6 3! +  8 4! − ⋯ for all . Using this method, it is easy to find the series up to the  10 or  20 terms. It can be shown that this is the Taylor Series for  − 2 . Example 1 Find the Taylor Series about  = 0 for  () = 1 1 +  2 . Solution The binomial series, or geometric series, tells us that 1 1 +  = (1 + ) −1 = 1 −  +  2 −  3 +  4 − ⋯ for −1 <  < 1. Substituting  =  2 gives 1 1 +  2 = 1 −  2 +  4 −  6 +  8 − ⋯ for −1 <  < 1, 10.3 FINDING AND USING Taylor Series 531 which is the Taylor Series for 1 1 +  2 . Notice that substitution can affect the radius of convergence.

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  Calculus: Single and Multivariable

  • Deborah Hughes-Hallett, William G. McCallum, Andrew M. Gleason, Eric Connally, Daniel E. Flath, Selin Kalaycioglu, Brigitte Lahme, Patti Frazer Lock, David O. Lomen, David Lovelock, Guadalupe I. Lozano, Jerry Morris, David Mumford, Brad G. Osgood, Cody L. Patterson, Douglas Quinney, Karen R. Rhea, Ayse Arzu Sahin, Ad(Authors)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  Assuming the function has all its derivatives defined, finding the coefficients can always be done, in theory at least, by differentiation. That is how we derived the four most important Taylor Series, those for the functions   , sin , cos , and (1 + )  . For many functions, however, computing Taylor Series coefficients by differentiation can be a very laborious business. We now introduce easier ways of finding Taylor Series, if the series we want is closely related to a series that we already know. New Series by Substitution Suppose we want to find the Taylor Series for  − 2 about  = 0. We could find the coefficients by differentiation. Differentiating  − 2 by the chain rule gives −2 − 2 , and differentiating again gives −2 − 2 + 4 2  − 2 . Each time we differentiate we use the product rule, and the number of terms grows. Finding the tenth or twentieth derivative of  − 2 , and thus the series for  − 2 up to the  10 or  20 terms, by this method is tiresome (at least without a computer or calculator that can differentiate). Fortunately, there’s a quicker way. Recall that   = 1 +  +  2 2! +  3 3! +  4 4! + ⋯ for all . Substituting  = − 2 tells us that  − 2 = 1 + (− 2 ) + (− 2 ) 2 2! + (− 2 ) 3 3! + (− 2 ) 4 4! + ⋯ = 1 −  2 +  4 2! −  6 3! +  8 4! − ⋯ for all . Using this method, it is easy to find the series up to the  10 or  20 terms. It can be shown that this is the Taylor Series for  − 2 . Example 1 Find the Taylor Series about  = 0 for  () = 1 1 +  2 . Solution The binomial series, or geometric series, tells us that 1 1 +  = (1 + ) −1 = 1 −  +  2 −  3 +  4 − ⋯ for −1 <  < 1. Substituting  =  2 gives 1 1 +  2 = 1 −  2 +  4 −  6 +  8 − ⋯ for −1 <  < 1, 10.3 FINDING AND USING Taylor Series 531 which is the Taylor Series for 1 1 +  2 . Notice that substitution can affect the radius of convergence.

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  Real and Complex Analysis

  • Christopher Apelian, Steve Surace(Authors)
  • 2009(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  We now formally define this unique power series associated with f at ξ 0 as the Taylor Series of f centered at ξ 0 . Definition 2.1 Suppose f : D → F is infinitely differentiable at ξ 0 ∈ D ⊂ F . Then the Taylor Series of f centered at ξ 0 is defined to be the power series given by ∞ j =0 f ( j ) ( ξ 0 ) j ! ( ξ -ξ 0 ) j . (9.13) The above definition says nothing about where the Taylor Series of f cen-tered at ξ 0 converges. Of course, it must converge at ξ = ξ 0 (to f ( ξ 0 ) in fact), but where else might it converge? If it converges at some other ξ ∈ D , is it necessarily true that it converges to the value f ( ξ ) there? Taylor’s theorem answers these questions. Since there are significant differences in the real and complex cases, we will state Taylor’s theorem separately for each. In the real case, we can appeal to Taylor’s theorem with remainder from Chapter 6. Recall from this result that if f : N r ( x 0 ) → R has n + 1 deriva-tives on N r ( x 0 ), then f ( x ) = n j =0 f ( j ) ( x 0 ) j ! ( x -x 0 ) j + R n ( x ) on N r ( x 0 ), (9.14) where R n ( x ) ≡ f ( n +1) ( c ) ( n +1)! ( x -x 0 ) n +1 for some c between x 0 and x . Suppose in addition we know that f is C ∞ on N r ( x 0 ) (not just at x 0 ) and that lim n →∞ R n ( x ) = 0 on N r ( x 0 ). Then, taking the limit as n → ∞ in (9.14) obtains f ( x ) = ∞ j =0 f ( j ) ( x 0 ) j ! ( x -x 0 ) j on N r ( x 0 ), i.e., the Taylor Series of f centered at x 0 converges at each x ∈ N r ( x 0 ), and it converges to the value f ( x ). This is the basis for referring to a Taylor Series Taylor Series 475 expansion of a function f centered at a point ξ 0 as a Taylor Series representation of f centered at that point. We have established the following result. Theorem 2.2 (Taylor’s Theorem for Real Functions) Let f : D 1 → R be C ∞ on some neighborhood N r ( x 0 ) ⊂ D 1 , and suppose lim n →∞ R n ( x ) = 0 on N r ( x 0 ).

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  Single Variable Calculus

  Volume 1: Single Variable Calculus

  • Galina Filipuk, Andrzej Kozłowski(Authors)
  • 2019(Publication Date)
  • De Gruyter

    (Publisher)

  6  Sequences and series of functions

  In this chapter we will consider the problem of approximating functions by polynomials and representing functions by power series. We will discuss the Taylor Series and Mathematica ® ’s function Series . Next we consider uniform and almost uniform convergence of function sequences and series and conditions for continuity and differentiability of their limits and sums.

  6.1  Power series continued

  We will now return to the subject of power series, which we already introduced in Section 3.10 .

  A power series can be viewed as a generalization of a polynomial. Recall that polynomials are just lists of numbers

  (

  a 0

  , … ,

  a d

  )

  together with rules for adding and multiplying any two such lists. It is convenient to write polynomials in the form

  a 0

  +

  a 1

  x +

  a 2

  x 2

  + … +

  a n

  x n

  , where x is called an indeterminate or a variable. Formal power series are defined in exactly the same way, except that we consider infinite rather than just finite sequences. Such a series can be written in the form

  a 0

  +

  a 1

  x +

  a 2

  x 2

  + … +

  a n

  x n

  + …

  , where x is again a variable. Two such series can also be added and multiplied (the multiplication being given by the Cauchy product).

  One important difference between polynomials and formal power series is that a polynomial always defines a function given by substituting numbers for the variable x. However, in the case of formal power series the situation is more complicated for although we can “substitute” a number for x, the number series thus obtained may not be convergent (it is always convergent when we substitute 0). To obtain a function we need to find the set of points where the series is convergent.

  Also recall from Section 3.10 that we consider power series of the form

  ∑

  n = 0

  ∞

  a n

  ( x −

  x 0

  )

  n

  , where

  x 0

  is called the center of the series. It is not hard to prove that a power series defines a function that is continuous in the interior of its region of convergence. A result by Abel known as Abel’s continuity/limit theorem [14

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  The Calculus Lifesaver

  All the Tools You Need to Excel at Calculus

  • Adrian Banner(Author)
  • 2009(Publication Date)
  • Princeton University Press

    (Publisher)

  The coefficients of this power series are given by a n = f ( n ) ( a ) /n !. The series is called the Taylor Series of f about x = a . So, starting with a function, we have defined a power series. Take a closer look at the definition of the Taylor Series above. It should look familiar. In fact, the formula is very similar to the definition of the Taylor polynomial P N ( x ) from Section 24.1.3 above. The only difference is that the sum doesn’t stop at n = N : it keeps on going to ∞ . In other words, the Taylor polynomial P N ( x ) is the N th partial sum of the Taylor Series. We’ll explore the connection between Taylor polynomials and Taylor Series in the next section. First, we have just one more definition: the Maclaurin series of f is just another name for the Taylor Series of f about x = 0. So it’s given by ∞ X n =0 f ( n ) (0) n ! x n , or in expanded form by f (0) + f 0 (0) x + f 00 (0) 2! x 2 + f (3) (0) 3! x 3 + f (4) (0) 4! x 4 + · · · . Whenever you see the words “Maclaurin series,” mentally replace them by “Taylor Series with a = 0” and you’ll do just fine. 24.2.3 Convergence of Taylor Series OK, let’s review the situation. We started out with a function f and a number a , and we constructed the Taylor Series of f about x = a : ∞ X n =0 f ( n ) ( a ) n ! ( x -a ) n . This is a power series with center a , but it’s not just any old power series: it encapsulates the values of all the derivatives of f at x = a . It would be really cool if we could write f ( x ) = ∞ X n =0 f ( n ) ( a ) n ! ( x -a ) n , since then we’d know that the Taylor Series converges for any x and also that it converges to the original function value f ( x ). The problem is, the above equation isn’t always valid. The series could diverge for some values of x , or even every value of x (except x = a : as we’ve seen, a power series always converges at its center).

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  The Art and Craft of Problem Solving

  • Paul Zeitz(Author)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  9 Calculus In this chapter, we take it for granted that you are familiar with the basic calculus ideas such as limits, continuity, differentiation, integration, and power series. On the other hand, we assume that you may have heard of, but not mastered: ∙ Formal “ − ” proofs ∙ Taylor Series with “remainder” ∙ The mean value theorem In contrast to, say, Chapter 7, this chapter is not a systematic, self-contained treatment. Instead, we concentrate on just a few important ideas that enhance your understanding of how calculus works. Our goal is twofold: to uncover the practical meaning of some of the things that you have already studied, by developing useful reformulations of old ideas; and to enhance your intuitive understanding of calculus, by showing you some useful albeit non-rigorous “moving curtains.” The meaning of this last phrase is best understood with an example. 9.1 The Fundamental Theorem of Calculus To understand what a moving curtain is, we shall explore, in some detail, the most important idea of elementary calculus. This example also introduces a number of ideas that we will keep returning to throughout the chapter. Example 9.1.1 What is the fundamental theorem of calculus (FTC), what does it mean, and why is it true? Partial Solution: You have undoubtedly learned about the FTC. One formulation of it says that if  is a continuous function, 1 then ∫    () =  () −  (), (9.1.1) where  is any antiderivative of  ; i.e.,  ′ () =  (). This is a remarkable statement. The left-hand side of equation (9.1.1) can be interpreted as the area bounded by the graph of  =  (), the -axis, and the vertical lines  =  and  = . The right-hand side has a completely different meaning, since 1 In this chapter, we will assume that the domain and range of all functions are subsets of the real numbers.

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