Explanation Given: The function f ( x ) = sinh x + cosh x , a = 0 . Theorem used: Taylor’s remainder theorem “The remainder of the Taylor series is R n ( x ) = f ( n + 1 ) ( c ) ( n + 1 ) ! ( x − a ) n + 1 for some point c between x and a .” Calculation: Obtain the remainder in the Taylor series for f ( x ) = sinh x + cosh x centered at 0. Substitute 0 for a in R n ( x ) = f ( n + 1 ) ( c ) ( n + 1 ) ! ( x − a ) n + 1 , R n ( x ) = f ( n + 1 ) ( c ) ( n + 1 ) ! ( x − a ) n + 1 = f ( n + 1 ) ( c ) ( n + 1 ) ! ( x − 0 ) n + 1 = f ( n + 1 ) ( c ) ( n + 1 ) ! ( x ) n + 1 Therefore, the remainder in the Taylor series is R n ( x ) = f ( n + 1 ) ( c ) ( n + 1 ) ! ( x ) n + 1 for some c . To show that lim n → ∞ R n ( x ) = 0 for all x in the interval of convergence. The interval of convergence for Taylor series f ( x ) = sinh x + cosh x is ( − ∞ , ∞ )

Calculus: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134763644

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

Publisher: PEARSON

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Chapter 11, Problem 47RE

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To write: The remainder term for the Taylor series for given function centered at a, also show that limn→∞Rn(x)=0 for all x in the interval of convergence.

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Chapter 11, Problem 46RE

Chapter 11, Problem 48RE

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Chapter 11 Solutions

Calculus: Early Transcendentals (3rd Edition)

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To write: The remainder term for the Taylor series for given function centered at a, also show that limn→∞Rn(x)=0 for all x in the interval of convergence.

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