University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 9 - Section 9.8 - Taylor and Maclaurin Series - Exercises - Page 536: 37



Work Step by Step

The Taylor's series for $f(x)$ at $x=a$ is as follows: $\Sigma_{n=0}^{\infty} \dfrac{f^{k}(a)}{k!}(x-a)^k=f(a)+f'(a)(x-a)+.....+\dfrac{f^{n}(a)}{n!}(x-a)^n$ Therefore, $f(x)=e^x$ and $f^n(a)=e^a$ $e^ x=e^a+e^a(x-a) +e^a \dfrac{(x-a)^2}{2!}+......$ or, $e^x=e^a[1+(x-a)+\dfrac{(x-a)^2}{2!}+...+\dfrac{(x-a)^n}{n!}]$
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