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: 27


$e^x\geq 1+x$ for all x

Work Step by Step

The Taylor series for $e^{x}$ with $r$ in $(-\infty,\infty)$ is $e^x=1+x+\frac{1}{2!}x^2+\frac{1}{3!}x^3+...+\frac{1}{n!}x^n+...$ $e^x=(1+x)+(\frac{x^2}{2!}+\frac{x^3}{3!})+...+\frac{1}{n!}x^n+...$ $e^x-(1+x)\geq0$ (0 represents $(\frac{x^2}{2!}+\frac{x^3}{3!}+...+\frac{1}{n!}x^n+...)$ $e^x\geq 1+x$ Hence, we can suggest that $e^x\geq 1+x$ for all x
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