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Unformatted text preview: TAYLOR NOTES Math 126: Taylor Polynomials and Infinite Series (An Epilogue to Single Variable Calculus) K. P. Bube University of Washington, Department of Mathematics, Seattle, WA 98195 March 2005 Introduction In this Epilogue to Single Variable Calculus, we introduce two very useful concepts that are closely related to topics we have studied in Math 124 and Math 125. These concepts are used widely throughout mathematics, the sciences, and engineering. They are discussed in more detail in Chapter 11 in our Math 126 text ( Calculus: Early Transcendentals, 5th Edition by James Stewart), and they are covered in depth in Math 327 and Math 328. In Math 126, we will not go into as much detail as does Chapter 11 in Stewart. The goal of these notes is to give you an introduction to these concepts, with an emphasis on aspects that are directly related to the calculus you have already studied. The first topic is an introduction to Taylor polynomials. These are an extension of the concept of a tangent line. Taylor polynomials are higher-degree polynomials that give very good approximations to a function near a point x = a . The second topic is infinite series. Infinite series are closely related to improper integrals. Recall that the value of an improper integral is defined by Z a f ( x ) dx = lim t Z t a f ( x ) dx, the limit of definite integrals on finite intervals. Similarly, we define the value of an infinite series to mean the limit of sums with a finite number of terms: X k =1 a k = lim n n X k =1 a k ! . Not only is the definition of infinite series similar to the definition of improper integrals, but also one of the important tools for understanding infinite series, called the Integral Test, involves comparing infinite series with improper integrals. Part 3 of these notes brings these two topics together and considers the limit of the Taylor polynomials as the degree n goes to infinity. Preliminary: Sigma Notation A useful shorthand for writing sums, called sigma notation , uses a X , a capital Greek letter sigma (corresponding to S for sum). For integers m and n with m n , the notation n X k = m a k represents the sum n X k = m a k = a m + a m +1 + a m +2 + + a n- 1 + a n . For example, 7 X k =2 k 2 = 2 2 + 3 2 + 4 2 + 5 2 + 6 2 + 7 2 = 139 . To know what the sum n X k = m a k is, we need to know the value of each term a k for all integers k satisfying m k n : if we know all of these values, we could then list them ( a m , a m +1 , a m +2 , . . . , a n- 1 , a n ) and add them up. Notice the great similarities between sums using the sigma notation and definite integrals Z b a f ( x ) dx : the dummy index k in the sum plays the role of the dummy variable x in the definite integral. The lower index m plays the role of a and the upper index n plays the role of b . For the sum, the dummy variable k ranges over all integer values satisfying m k n ; for the integral, the dummy variable...
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