Taylor Theorem - Contents 14 Taylor’s Theorem 197 14.1...

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Unformatted text preview: Contents 14 Taylor’s Theorem 197 14.1 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 14.1.1 Polynomials with Specific Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 14.2 Successive Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 14.3 Taylor Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 14.4 Properties of Taylor Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 14.5 Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 14.5.1 Error Term E n,a ( x ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 14.6 Approximations by Taylor Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 14.6.1 Error Estimate for f (- 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 14.6.2 Error Estimate for f (0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 14.6.3 Error Estimate for f (- 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 14.7 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 14.7.1 Ratio Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 14.7.2 Root Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 14.7.3 The Integral Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 14.7.4 Alternating Series Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 14.7.5 Alternating Series Approximation Theorem . . . . . . . . . . . . . . . . . . . . . . . . 216 14.7.6 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 14.7.7 Properties of Functions Represented by Power Series . . . . . . . . . . . . . . . . . . . 219 2 CONTENTS Chapter 14 Taylor’s Theorem 14.1 Polynomials Every polynomial can be expressed not only in powers of x but also in powers of x- a . Example : Write x 2 + x + 2 in powers of x- 1. Method 1: Set w = x- 1. Then x = w + 1 and x 2 + x + 2 = ( w + 1) 2 + ( w + 1) + 2 = w 2 + 3 w + 4. Therefore x 2 + x + 2 = ( x- 1) 2 + 3( x- 1) + 4. This method is not practical if the degree of the polynomial is greater than 3. Method 2: Write 2 + x + x 2 = C + C 1 ( x- 1) + C 2 ( x- 1) 2 . ( * ) Determine the coefficients as follows: 1. Evaluate ( * ) at x = 1 to obtain C = 4. 2. Differentiate both sides of ( * ): 1 + 2 x = C 1 + 2 C 2 ( x- 1). ( ** ) 3. Evaluate ( ** ) at x = 1 to get C 1 = 3....
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Taylor Theorem - Contents 14 Taylor’s Theorem 197 14.1...

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