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15. (a) Use the Taylor/Maclaurin theorem to show that the power series for the expo- nential function is, for all real x, n! n=0 (b) Use (a) to find a power series for the function f(x) = e¯a°.

15. (a) Use the Taylor/Maclaurin theorem to show that the power series for the expo- nential function is, for all real x, n! n=0 (b) Use (a) to find a power series for the function f(x) = e¯a°.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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15. (a) Use the Taylor/Maclaurin theorem to show that the power series for the expo- nential function is, for all real x, n! n=0 (b) Use (a) to find a power series for the function f(x) = e-². (c) By integrating the power series in (b) determine an alternating series formula for the definite integral r1/2 dx, and use the alternating series theorem to compute this integral (which cannot be done with antiderivatives, by the way) correct to 2 decimal places.
Transcribed Image Text:15. (a) Use the Taylor/Maclaurin theorem to show that the power series for the expo- nential function is, for all real x, n! n=0 (b) Use (a) to find a power series for the function f(x) = e-². (c) By integrating the power series in (b) determine an alternating series formula for the definite integral r1/2 dx, and use the alternating series theorem to compute this integral (which cannot be done with antiderivatives, by the way) correct to 2 decimal places.
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