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: 5


diverges for all $x$

Work Step by Step

Given $$\sum_{n=0}^{\infty} n^{n} x^{n}$$ Since $a_n = n^{n} x^{n}$ and $a_{n+1} = (n+1)^{n+1} x^{n+1}$, then \begin{aligned} \rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\ &=\lim _{n \rightarrow \infty}\left|\frac{(n+1)^{n+1} x^{n+1}}{n^{n} x^{n}}\right|\\ &=\lim _{n \rightarrow \infty}\left|x \left(\frac{n+1}{n}\right)^n (n+1 )\right|\\ &=\lim _{n \rightarrow \infty}\left|x \left(1+\frac{1}{n}\right)^n (n+1 )\right|\\ &=|x|\lim _{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^n\lim _{n \rightarrow \infty} (n+1 ) \\ &=\infty \end{aligned} Then the series diverges for all $x$
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