Calculus (3rd Edition)

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

Chapter 11 - Infinite Series - 11.6 Power Series - Exercises - Page 578: 30


$$ -\infty\lt x\lt \infty $$

Work Step by Step

Given $$\sum_{n=0}^{\infty} \frac{(x-4)^{n}}{n !}$$ Since $a_n = \frac{(x-4)^{n}}{n !}$ and $a_{n+1} = \frac{(x-4)^{n+1}}{(n+1) !}$, then \begin{aligned} \rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\ & =\lim _{n \rightarrow \infty}\left|\frac{\frac{(x-4) (x-4)^{n}}{(n+1) n !}}{\frac{(x-4)^{n}}{n !}}\right|\\ & =\lim _{n \rightarrow \infty}\left|(x-4) \frac{1}{n+1}\right|\\ & =|(x-4)| \lim _{n \rightarrow \infty} \frac{1}{n+1}\\ &=0 \end{aligned} Then the interval of convergence is $$ -\infty\lt x\lt \infty $$
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