Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 12 - Sequences and Series - Chapter Review - Review Exercises - Page 662: 50


For $r$ in $(-\infty,\infty)$, the Taylor series for $f(x)=x^6e^{-x}$ is $x^6-x^7-\frac{1}{2}x^8-\frac{1}{6}x^9...+\frac{(-x)^{n+6}}{n!}+...$

Work Step by Step

We are given $f(x)=x^6e^{-x}$ for $r$ in $(-\infty,\infty)$ The Taylor series for $e^x$ is $1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...+\frac{1}{n!}x^n+...$ With $c=x^6$, the Taylor series for $f(x)=x^6e^{-x}$ is $x^6+x^6.(-x)+x^6\frac{(-x)^2}{2!}+x^6\frac{(-x)^3}{3!}+...+x^6\frac{(-x)^n}{n!}+...$ $=x^6-x^7-\frac{1}{2}x^8-\frac{1}{6}x^9...+\frac{(-x)^{n+6}}{n!}+...$
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