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: 26


$e^{-x}\approx1-x+\frac{x^2}{2}$ for all x close to zero

Work Step by Step

The Taylor series for $e^{-x}$ with $r$ in $(-\infty,\infty)$ is $1+(-x)+\frac{1}{2!}(-x)^2+\frac{1}{3!}(-x)^3+...+\frac{1}{n!}(-x)^n+...$ $=1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+...+\frac{1}{n!}(-x)^n+...$ We can use the first few terms of a Taylor Series to get an approximate value for a function. Hence, we can suggest that $e^{-x}\approx1-x+\frac{x^2}{2}$ for all x close to zero
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