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b) i. Use the Maclaurin series for e to find an power series expression for e-1/2x² dx. ii. Hence evaluate S dx as an infinite series. Use the ratio test to prove whether this series converges or not. (You do not have to evaluate the sum if it converges.) e-x² iii. Consider the integral I = L e ²dx. By writing I² as a double integral over vari- ables (x, y), or otherwise, use a suitable coordinate transformation to evaluate this integral, and hence evaluate I.

b) i. Use the Maclaurin series for e to find an power series expression for e-1/2x² dx. ii. Hence evaluate S dx as an infinite series. Use the ratio test to prove whether this series converges or not. (You do not have to evaluate the sum if it converges.) e-x² iii. Consider the integral I = L e ²dx. By writing I² as a double integral over vari- ables (x, y), or otherwise, use a suitable coordinate transformation to evaluate this integral, and hence evaluate I.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 68E
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b) i. Use the Maclaurin series for e to find an power series expression for e-1/2x² dx. ii. Hence evaluate C' e-² dx as an infinite series. Use the ratio test to prove whether this series converges or not. (You do not have to evaluate the sum if it converges.) iii. Consider the integral I = L e -² dx. By writing I² as a double integral over vari- ables (x, y), or otherwise, use a suitable coordinate transformation to evaluate this integral, and hence evaluate I.
Transcribed Image Text:b) i. Use the Maclaurin series for e to find an power series expression for e-1/2x² dx. ii. Hence evaluate C' e-² dx as an infinite series. Use the ratio test to prove whether this series converges or not. (You do not have to evaluate the sum if it converges.) iii. Consider the integral I = L e -² dx. By writing I² as a double integral over vari- ables (x, y), or otherwise, use a suitable coordinate transformation to evaluate this integral, and hence evaluate I.
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