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Get Full Access to Calculus: Early Transcendentals - 1 Edition - Chapter 9.3 - Problem 80ae
Get Full Access to Calculus: Early Transcendentals - 1 Edition - Chapter 9.3 - Problem 80ae

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# Nonconvergence to f Consider the function a. Use the

ISBN: 9780321570567 2

## Solution for problem 80AE Chapter 9.3

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 $$x \neq 0$$.

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