Explanation Proof: Result used: 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. Given: The radius of convergence of the power series f ( x ) = ∑ k = 0 ∞ c k x k is R . Calculation: Consider the power series f ( x ) = ∑ k = 0 ∞ c k x k . That is, lim k → ∞ | a k + 1 a k | = lim k → ∞ | c k + 1 x k + 1 c k x k | (1) Multiply the power series by x m on both sides, x m f ( x ) = x m ∑ k = 0 ∞ c k x k = ∑ k = 0 ∞ c k x k + m That is, x m f ( x ) = ∑ k = 0 ∞ c k x k + m

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 73E

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To prove: The radius of convergence of the power series xmf(x) is R.

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Chapter 9.2, Problem 72E

Chapter 9.2, Problem 74E

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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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To prove: The radius of convergence of the power series x m f ( x ) is R .

Solution Summary: The author proves that the radius of convergence of the power series xmf(x-) is R.

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To prove: The radius of convergence of the power series xmf(x) is R.

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Chapter 9 Solutions