Explanation Given: The function f ( x ) = cos 2 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 centered and 0. Substitute 0 for a in R n ( x ) = f ( n + 1 ) ( c ) ( n + 1 ) ! ( x − a ) n + 1 , By the theorem stated above, the remainder is 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 cos 2 x is ( − ∞ , ∞ ) . Since the ( n + 1 ) th derivative of cos ( 2 x ) is 2 n + 1 times either ± cos 2 x or ± sin 2 x , | f ( n + 1 ) ( c ) | ≤ 2 n + 1

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

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ISBN: 9780134996103

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

Publisher: PEARSON

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Chapter 11.3, Problem 64E

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To find: The remainder in the in the Taylor series centered at a; show that limn→∞Rn(x)=0 for all x in the interval of convergence.

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Chapter 11.3, Problem 63E

Chapter 11.3, Problem 65E

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Chapter 11 Solutions

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (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 remainder in the in the Taylor series centered at a ; show that lim n → ∞ R n ( x ) = 0 for all x in the interval of convergence.

Solution Summary: The author calculates the remainder in the Taylor series centered at a.

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To find: The remainder in the in the Taylor series centered at a; show that limn→∞Rn(x)=0 for all x in the interval of convergence.

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Chapter 11 Solutions