a. Explanation Result used: The Taylor series representation of − ln ( 1 − x ) = x + x 2 2 + x 3 3 + ⋯ + x k k + ⋯ = ∑ k = 1 ∞ x k k , for − 1 ≤ x < 1 . Given: The function f ( x ) = − ln ( 1 − x ) . Calculation: Differentiate the function f ( x ) = − ln ( 1 − x ) term by term as follows, f ′ ( x ) = d d x ( x + x 2 2 + x 3 3 + ⋯ ) = d d b. To determine To identify: The function represented by the differentiated series. c. To determine To give: The interval of convergence of the differentiated series f ′ ( x ) = 1 1 − x .

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 11.4, Problem 32E

To determine

To find: The derivative of the Taylor series about 0.

To determine

To identify: The function represented by the differentiated series.

To determine

To give: The interval of convergence of the differentiated series f′(x)=11−x.

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Chapter 11.4, Problem 31E

Chapter 11.4, Problem 33E

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K Consider the function f(x)=sin(x7). a. Differentiate the Taylor series afout 0 for f(x). b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative. a. Choose the correct answer below. OA. 7x6. 7x20 2! O C. -7x¹3+ 7x34 4! 7x27 3! 7x 48 6! 7x41 5! + ... О) в. 7x6 - 7x20 2! OD. -7x¹ 13 b. f'(x) = c. The interval of convergence of the power series for the derivative is. (Simplify your answer. Type an inequality or a compound inequality. Type an expression using x as the variable.) + 7x34 4! 7x48 6! 7x27 7x4 41 3! 5! + ...

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 derivative of the Taylor series about 0.

Solution Summary: The author explains how to find the derivative of the Taylor series, which is fprime (x)=-mathrm

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To find: The derivative of the Taylor series about 0.

To identify: The function represented by the differentiated series.

To give: The interval of convergence of the differentiated series f′(x)=11−x.

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Chapter 11 Solutions