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 577: 6



Work Step by Step

We apply the ratio test $$ \rho=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\lim _{n \rightarrow \infty} |\frac{ x^{n+1} (n+1)! }{ x^{n} n! }|= |x|\lim _{n \rightarrow \infty} n+1 $$ Hence, the series $\Sigma_{n=0}^{\infty} x^{n} n!$ converges if and only if $\rho= |x|\lim _{n \rightarrow \infty} n+1 \lt1$, which is true only for the values $x\lt 0$.
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