Calculus with Applications (10th Edition)

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

Chapter 12 - Sequences and Series - 12.5 Taylor Series - 12.5 Exercises - Page 647: 16



Work Step by Step

We are given $f(x)=x^4e^{2x}$ for $r$ in $(-\infty,\infty)$ The Taylor series for $e^x$ is $1+x+\frac{1}{2!}x^2+\frac{1}{3!}x^3+...+\frac{1}{n!}x^n+...$ The Taylor series for $f(x)=x^4e^{2x}$ is $f(x)=x^4.1+x^4(2x)+x^4\frac{1}{2!}(2x)^2+x^4\frac{1}{3!}(2x)^3+...+x^4\frac{1}{n!}(2x)^n+...$ $=x^4+2x^5+2x^6+\frac{4x^7}{3}+...+\frac{(2x)^{n+4}}{n!}+...$
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