University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 9 - Section 9.5 - Absolute Convergence; The Ratio and Root Tests - Exercises - Page 515: 5

Answer

Converges

Work Step by Step

Consider $a_n=\dfrac{n^4}{(-4)^n}$ and $|a_n|=|\dfrac{n^4}{(-4)^n}|=\dfrac{n^4}{4^n}$ Now, $l=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{(n+1)^4}{4^{n+1}}}{\dfrac{n^4}{4^n}}|$ Thus, we have $l=\lim\limits_{n \to \infty}|\dfrac{(n+1)^4}{4n^4}|=\dfrac{1}{4} \lt 1$ Hence, the series Converges absolutely by the ratio test.
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