Explanation Result used: (1) Ratio test : Let ∑ k = 1 ∞ a k be an infinite series and let r = lim k → ∞ | a k + 1 a k | . If r < 1 , then the series ∑ k = 1 ∞ a k is converges. (2) Geometric series ∑ k = 0 ∞ r n is converges if | r | < 1 and diverges if | r | ≥ 1 . (3) The power series representation of 1 1 − x = ∑ k = 0 ∞ x k . Given: The function is f ( x ) = 1 1 + x 2 . Calculation: The power series representation for f ( x ) = 1 1 + x 2 is obtained as follows, Substitute − x 2 for x in 1 1 − x = ∑ k = 0 ∞ x k , 1 1 + x 2 = ∑ k = 0 ∞ ( − x 2 ) k = ∑ k = 0 ∞ ( − 1 ) k ( x ) 2 k Therefore, the power series representation for f ( x ) = ∑ k = 0 ∞ ( − 1 ) k ( x ) 2 k . Let a k = ( − 1 ) k ( x ) 2 k . Then, a k + 1 = ( − 1 ) k + 1 ( x ) 2 ( k + 1 ) . Use the Ratio test to compute the value | a k + 1 a k | . | a k + 1 a k | = | ( − 1 ) k + 1 ( x ) 2 ( k + 1 ) ( − 1 ) k ( x ) 2 k | Take lim k → ∞ on both sides

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

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

Publisher: PEARSON

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Chapter 9.2, Problem 47E

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To find: The power series representations centered at 0 and find the interval of convergence for the resulting series.

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Chapter 9.2, Problem 46E

Chapter 9.2, Problem 48E

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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 power series representations centered at 0 and find the interval of convergence for the resulting series.

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Chapter 9 Solutions