Calculus (3rd Edition)

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

Chapter 11 - Infinite Series - 11.5 The Ratio and Root Tests and Strategies for Choosing Tests - Exercises - Page 568: 26


Diverges for all $k$

Work Step by Step

Given $$\sum_{n=1}^{\infty} \frac{2^n}{n^k}$$ By using the Ratio test, we get \begin{align*} \rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\ &=\lim _{n \rightarrow \infty}\left| \frac{2^{n+1}}{(n+1)^k}\frac{n^k}{2^n}\right|\\ &= 2\lim _{n \rightarrow \infty} |\frac{n^k}{(n+1)^k}|\\\ &=2 \end{align*} Thus the series diverges for all $k$.
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