Explanation Result used: The power series representation of 1 1 + x = ∑ k = 0 ∞ x k . Given: The given series ∑ k = 2 ∞ k ( k − 1 ) x k 3 k . Calculation: Obtain the function represented power series. ∑ k = 2 ∞ k ( k − 1 ) x k 3 k = x 2 ∑ k = 2 ∞ k ( k − 1 ) x k − 2 3 k = x 2 d 2 d x 2 ( ∑ k = 2 ∞ ( x 3 ) k ) = x 2 d 2 d x 2 ( x 2 3 2 + x 3 3 3 + x 4 3 4 + ⋯ ) = x 2 d 2 d x 2 ( x 2 3 2 ( 1 + x 3 + x 2 3 2 + ⋯ ) ) Use the Taylor series ∑ k = 1 ∞ ( x 3 ) k = 1 1 − x 3 in the above, ∑ k = 2 ∞ k ( k − 1 ) x k 3 k = x 2 d 2 d x 2 ( x 2 9 ( 3 3 − x ) ) = x 2 d 2 d x 2 ( x 2 9 − 3 x ) = x 2 d d x ( d d x ( x 2 ) ⋅ ( 9 − 3 x ) − d d x ( 9 − 3 x ) ⋅ ( x 2 ) ( 9 − 3 x ) 2 ) = x 2 d d x ( 2 x ( 9 − 3 x ) − ( − 3 )

Calculus, Single Variable: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134766850

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

Publisher: PEARSON

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Chapter 11.4, Problem 63E

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To identify: The functions represented by the given power series.

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Chapter 11.4, Problem 62E

Chapter 11.4, Problem 64E

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Calculus, Single Variable: Early Transcendentals (3rd Edition)

Ch. 11.1 - Verify that p3 satisfies p3(k)(a)=f(k)(a), for k =...Ch. 11.1 - Prob. 2QC Ch. 11.1 - Prob. 3QC Ch. 11.1 - Write out the next two Taylor polynomials p4 and...Ch. 11.1 - Prob. 5QC Ch. 11.1 - Prob. 6QC Ch. 11.1 - Suppose you use a second-order Taylor polynomial...Ch. 11.1 - Does the accuracy of an approximation given by a...Ch. 11.1 - The first three Taylor polynomials for f(x)=1+x...Ch. 11.1 - Prob. 4E

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The functions represented by the given power series.

Solution Summary: The author identifies the functions represented by the given power series: -6x2(x-3)3.

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Chapter 11 Solutions