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Nonconvergence to f Consider the function a. Use the

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett ISBN: 9780321570567 2

Solution for problem 80AE Chapter 9.3

Calculus: Early Transcendentals | 1st Edition

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Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

Calculus: Early Transcendentals | 1st Edition

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Problem 80AE

Nonconvergence to f Consider the function

\(f(x)=\left\{\begin{array}{ll} e^{-1 / x^{2}} & \text { if } x \neq 0 \\ 0 & \text { if } x=0 \end{array}\right.\)

a. Use the definition of the derivative to show that \(f^{\prime}(0)=0\).

b. Assume the fact that \(f^{(k)}(0)=0\), for k = 1, 2, 3, .... (You can write a proof using the definition of the derivative.) Write the Taylor series for f centered at 0.

c. Explain why the Taylor series for f does not converge to f for \(x \neq 0\).

Step-by-Step Solution:
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Chapter 9.3, Problem 80AE is Solved
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Textbook: Calculus: Early Transcendentals
Edition: 1
Author: William L. Briggs, Lyle Cochran, Bernard Gillett
ISBN: 9780321570567

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Nonconvergence to f Consider the function a. Use the