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 516: 25



Work Step by Step

Consider $|a_n|=(1-\dfrac{3}{n})^n$ Now, $\lim\limits_{n \to \infty} |a_n|=\lim\limits_{n \to \infty} (1-\dfrac{3}{n})^n$ Thus, we have $=\lim\limits_{n \to \infty} (1+(-\dfrac{3}{n}))^n$ So, $\lim\limits_{n \to \infty} |a_n|=e^{-3}=\dfrac{1}{e^3}$ We need $\lim\limits_{n \to \infty} |a_n| =0$ for the series to converge, but we can see that $\lim\limits_{n \to \infty} |a_n| \ne 0$ Hence, The Series Diverges.
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