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.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 67E
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