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: 21

Answer

Converges for all exponents $k$

Work Step by Step

Given $$\sum_{n=1}^{\infty} n^k3^{-n}$$ 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}\frac{ (n+1)^{k}}{3^{n+1} }\frac{3^n}{n^k} \\ &= \lim _{n \rightarrow \infty} \frac{ (n+1)^{k}}{3n^k}\\ &= \frac{1}{3}<1 \end{align*} Thus the series converges for all exponents $k$.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.