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Use the fact that e² = Σ k=0 k! 1+x+ (b) Find a series whose sum is · [₁₁-² 2! 3! (a) The antiderivatives of e-²/2 are not elementary functions; this means that, no matter how hard you try, you will not be able to evaluate [e-2²/2 da as a finite sum of functions we know. Instead, find a power series representation of the indefinite integral [e- -²/2 dr. For what a is your representation valid? +. -²/2 dx. for all a.

Use the fact that e² = Σ k=0 k! 1+x+ (b) Find a series whose sum is · [₁₁-² 2! 3! (a) The antiderivatives of e-²/2 are not elementary functions; this means that, no matter how hard you try, you will not be able to evaluate [e-2²/2 da as a finite sum of functions we know. Instead, find a power series representation of the indefinite integral [e- -²/2 dr. For what a is your representation valid? +. -²/2 dx. for all a.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Need help with this problem part a and b

Use the fact that \( e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \) for all \( x \). (a) The antiderivatives of \( e^{-x^2/2} \) are not elementary functions; this means that, no matter how hard you try, you will not be able to evaluate \( \int e^{-x^2/2} \, dx \) as a finite sum of functions we know. Instead, find a power series representation of the indefinite integral \( \int e^{-x^2/2} \, dx \). For what \( x \) is your representation valid? (b) Find a series whose sum is \( \int_0^1 e^{-x^2/2} \, dx \).
Transcribed Image Text:Use the fact that \( e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \) for all \( x \). (a) The antiderivatives of \( e^{-x^2/2} \) are not elementary functions; this means that, no matter how hard you try, you will not be able to evaluate \( \int e^{-x^2/2} \, dx \) as a finite sum of functions we know. Instead, find a power series representation of the indefinite integral \( \int e^{-x^2/2} \, dx \). For what \( x \) is your representation valid? (b) Find a series whose sum is \( \int_0^1 e^{-x^2/2} \, dx \).
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