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.

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A: See the detailed solution below.

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