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EXAMPLE 10 (a) Evaluate the integral below as an infinite series. fe-z² (b) Evaluate the integral below correct to within an error of 0.0001. foot € SOLUTION (a) First we find the Maclaurin series for f(x) = ex. Although it's possible to use the direct method, let's find it simply by replacing x with -x² in the series for e given in Table 1 in the book. Thus for all values of x, ex² = 1 ∞o (-x²)^ n=0 n! x2 1! 0.5 1.0³ 4 fex dx = (1- = C + X- Now we integrate term by term. = Σ (-1)^ n=0 x² 1! 2x 2(1) 2! 2n e-² dx = x- n! X 2! x6 3! + ... x5 (b) The fundamental Theorem of Calculus gives 5.2 7.3! x7 This series converges for all x because the original series for e -2 + x5 (-1) 2n+1 x² (-1)^ 2n + 1 n! 10.5 x²n+1 + ... 18. X converges for all x. ).n! 5 2! 7.3! - 1/24 + - 1/5376 + 1/110592 - ... - 1/24 + 1/5376 + 1/110592 0.4278 X The Alternating Series Estimation Theorem shows that the error involved in this approximation is less than 0.0001. Activate Windows Go to Settings to activate Windows.

EXAMPLE 10 (a) Evaluate the integral below as an infinite series. fe-z² (b) Evaluate the integral below correct to within an error of 0.0001. foot € SOLUTION (a) First we find the Maclaurin series for f(x) = ex. Although it's possible to use the direct method, let's find it simply by replacing x with -x² in the series for e given in Table 1 in the book. Thus for all values of x, ex² = 1 ∞o (-x²)^ n=0 n! x2 1! 0.5 1.0³ 4 fex dx = (1- = C + X- Now we integrate term by term. = Σ (-1)^ n=0 x² 1! 2x 2(1) 2! 2n e-² dx = x- n! X 2! x6 3! + ... x5 (b) The fundamental Theorem of Calculus gives 5.2 7.3! x7 This series converges for all x because the original series for e -2 + x5 (-1) 2n+1 x² (-1)^ 2n + 1 n! 10.5 x²n+1 + ... 18. X converges for all x. ).n! 5 2! 7.3! - 1/24 + - 1/5376 + 1/110592 - ... - 1/24 + 1/5376 + 1/110592 0.4278 X The Alternating Series Estimation Theorem shows that the error involved in this approximation is less than 0.0001. Activate Windows Go to Settings to activate Windows.

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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EXAMPLE 10 (a) Evaluate the integral below as an infinite series. Je-z² (b) Evaluate the integral below correct to within an error of 0.0001. 6.0³. SOLUTION (a) First we find the Maclaurin series for f(x) = ex. Although it's possible to use the direct method, let's find it simply by replacing x with -x² in the series for e* given in Table 1 in the book. Thus for all values of x, ex² = Σ = 1 ∞o (-x²)" n=0 n! 1! 4 Sex²dx = (1- = C + X- = Now we integrate term by term. = Σ (-1)^ n=0 x² 1! 2x³ 2(1) 2! e-² dx = [x - x2n n! 1/24 + 1/24 + X 2! + +6 3! +5 x² 5 2 7 3! + x7 (-1)^2n+1 (-1) x² n! 2n+1 This series converges for all x because the original series for e*² converges for all x. (b) The fundamental Theorem of Calculus gives 6.³ 2n + 1 + ... 10.5 X ).n! 5.2! 7.3! 1/5376 + 1/110592 - 1/5376 + 1/110592 0.4278 x The Alternating Series Estimation Theorem shows that the error involved in this approximation is less than 0.0001. Activate Windows Go to Settings to activate Windows.
Transcribed Image Text:EXAMPLE 10 (a) Evaluate the integral below as an infinite series. Je-z² (b) Evaluate the integral below correct to within an error of 0.0001. 6.0³. SOLUTION (a) First we find the Maclaurin series for f(x) = ex. Although it's possible to use the direct method, let's find it simply by replacing x with -x² in the series for e* given in Table 1 in the book. Thus for all values of x, ex² = Σ = 1 ∞o (-x²)" n=0 n! 1! 4 Sex²dx = (1- = C + X- = Now we integrate term by term. = Σ (-1)^ n=0 x² 1! 2x³ 2(1) 2! e-² dx = [x - x2n n! 1/24 + 1/24 + X 2! + +6 3! +5 x² 5 2 7 3! + x7 (-1)^2n+1 (-1) x² n! 2n+1 This series converges for all x because the original series for e*² converges for all x. (b) The fundamental Theorem of Calculus gives 6.³ 2n + 1 + ... 10.5 X ).n! 5.2! 7.3! 1/5376 + 1/110592 - 1/5376 + 1/110592 0.4278 x The Alternating Series Estimation Theorem shows that the error involved in this approximation is less than 0.0001. Activate Windows Go to Settings to activate Windows.
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