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: 14


The series converges for all $x$ in the interval $(-1,1)$.

Work Step by Step

We apply the ratio test: $$ \rho=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\lim _{n \rightarrow \infty} |\frac{ (n+1)^7x^{n+1} }{n^7x^{n}}|=|x|\lim _{n \rightarrow \infty} \frac{(n+1)^7}{n^7}=|x| $$ hence, the series $\Sigma_{n=8}^{\infty} n^7x^{n}$ converges if and only if $|x|\lt1$. That is, the interval of convergence is $(- 1,1)$. Now, we check the end points: At $x=-1$, then $\Sigma_{n=8}^{\infty} n^7x^{n}=\Sigma_{n=8}^{\infty} n^7(-1)^{n}$, diverges (Limit Test) At $x=1 $, then $\Sigma_{n=8}^{\infty} n^7x^{n}=\Sigma_{n=8}^{\infty} n^7 $, diverges (Limit Test) Hence, the series converges for all $x$ in the interval $(-1,1)$.
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