Explanation Concept Used: If f ( x ) = ∑ n = 0 ∞ c n ( x − a ) n then its term wise derivative is given by f ′ ( x ) = ∑ n = 1 ∞ n c n ( x − a ) n − 1 Given the power series f ( x ) = ∑ k = 0 ∞ ( − 1 ) k x 2 k + 1 ( 2 k + 1 ) ! , to find the derivative we proceed as follows f ′ ( x ) = d d x ( f ( x ) ) = d d x ( ∑ k = 0 ∞ ( − 1 ) k x 2 k + 1 ( 2 k + 1 ) ! ) = ∑ k = 0 ∞ ( − 1 ) k ( 2 k + 1 ) ! d d x ( x 2 k + 1 ) = ∑ k = 0 ∞ ( − 1 ) k ( 2 k + 1 ) ! ( 2 k + 1 ) x 2 k = ∑ k = 0 ∞ ( − 1 ) k ( 2 k ) ! x 2 k f ′ ( x ) = ∑ k = 0 ∞ ( − 1 ) k ( 2 k ) ! x 2 k , to find the second derivative we proceed as follows f &

6th Edition

ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 8.7, Problem 53PS

To determine

To show: that f′′(x)=−f(x) for the power series f(x)=∑k=0∞ (−1)kx 2k+1(2k+1)! .

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Chapter 8.7, Problem 52PS

Chapter 8.7, Problem 54PS

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Chapter 8 Solutions

Calculus

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that f ′ ′ ( x ) = − f ( x ) for the power series f ( x ) = ∑ k = 0 ∞ ( − 1 ) k x 2 k + 1 ( 2 k + 1 ) ! .

Solution Summary: The author explains that the power series f(x)=displaystyle undersetn=0overset

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To show: that f′′(x)=−f(x) for the power series f(x)=∑k=0∞ (−1)kx 2k+1(2k+1)! .

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Chapter 8 Solutions