Explanation Given: The function f ( x ) = 1 + x , a = 0 , − 1 2 ≤ x ≤ 1 2 . 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 ) = 1 + x centered and 0. The ( n + 1 ) th derivative of 1 + x is d n + 1 d x n + 1 ( 1 + x ) = ± 1 ⋅ 3 ⋅ 5 ⋯ ( 2 n − 1 ) 2 n + 1 ⋅ ( 1 + x ) ( 2 n + 1 ) 2 . 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 ∈ ( − 1 2 , 1 2 ) . 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 ) = 1 + x is − 1 2 < x < 1 2

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Chapter 9, Problem 46RE

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To write: The remainder 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 9, Problem 45RE

Chapter 9, Problem 47RE

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Chapter 9 Solutions

CODE/CALC ET 3-HOLE

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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The remainder 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 for f(x)=sqrt1+x

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To write: The remainder 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 9 Solutions