Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 11 - Infinite Series - 11.7 Taylor Series - Exercises - Page 588: 13


$$f(x)=e^{-2}e^{ x} =\frac{1}{e^2}+\frac{x}{e^2}+\frac{x^2}{2!e^2}+\frac{x^3}{3!e^2}+\frac{x^4}{4!e^2}+...$$ Convergent for any value of $x$.

Work Step by Step

By making use of Table 2, we have the Maclaurin series for $f(x)=e^x$ as follows $$f(x)=e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...$$ Now by the comparison with the function $f(x)= e^{ x-2}=e^{-2}e^x$, we have the Maclaurin series as follows $$f(x)=e^{-2}e^{ x}\\ =e^{-2}(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...)\\ =\frac{1}{e^2}+\frac{x}{e^2}+\frac{x^2}{2!e^2}+\frac{x^3}{3!e^2}+\frac{x^4}{4!e^2}+...$$ Moreover, again from Table 2, the Maclaurin series for $f(x)=e^x$ is convergent for any value of $x$. Hence, the Maclaurin series for $f(x)= e^{ x-2}$ is also convergent for any value of $x$.
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