Explanation Given: The function is f ( x ) = a 2 + x 2 . Theorem used: Suppose the power series ∑ c k x k and ∑ d k x k converges to f ( x ) and g ( x ) respectively, on an interval I . (1) (Sum and difference): The power series ∑ ( c k ± d k ) x k converges to f ( x ) ± g ( x ) on I . (2) (Multiplication by a power): Suppose m is an integer such that k + m ≥ 0 for all terms of the power series x m ∑ c k x k = ∑ c k x k + m . This series converges to x m f ( x ) for all x ≠ 0 in I . when x = 0 , the series converges to lim x → 0 x m f ( x ) . (3) (Composition): If h ( x ) = b x m , where m is an integer and b is a nonzero real number, the power series ∑ c k ( h ( x ) ) k converges to the composite function. f ( h ( x ) ) ,for all x such that h ( x ) is in I . Calculation: The first four nonzero terms of the Maclaurin series is computed as follows, f ( x ) = a 2 + x 2 = 1 + x 2 a 2 Substitute ( x 2 a 2 ) for x in the Maclaurin series 1 + x = 1 + x 2 − x 2 8 + x 3 16 − ⋯

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Chapter 9.3, Problem 49E

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To find: The first four nonzero terms of the Maclaurin series and interval of convergence by using the Maclaurin series 1+x=1+x2−x28+x316−⋯, for −1≤x≤1.

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Chapter 9.3, Problem 48E

Chapter 9.3, Problem 50E

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Chapter 9 Solutions

Calculus: Early Transcendentals (2nd Edition)

Ch. 9.1 - Suppose you use a second-order Taylor polynomial...Ch. 9.1 - Does the accuracy of an approximation given by a...Ch. 9.1 - The first three Taylor polynomials for f(x)=1+x...Ch. 9.1 - Prob. 4E Ch. 9.1 - How is the remainder Rn(x) in a Taylor polynomial...Ch. 9.1 - Explain how to estimate the remainder in an...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...

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The first four nonzero terms of the Maclaurin series and interval of convergence by using the Maclaurin series 1 + x = 1 + x 2 − x 2 8 + x 3 16 − ⋯ , for − 1 ≤ x ≤ 1 .

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To find: The first four nonzero terms of the Maclaurin series and interval of convergence by using the Maclaurin series 1+x=1+x2−x28+x316−⋯, for −1≤x≤1.

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Chapter 9 Solutions