Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 11 - Section 11.7 - Strategy for Testing Series - 11.7 Exercises - Page 746: 25



Work Step by Step

Use the Ratio Test: $\lim_\limits{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$ Let $$a_n=\frac{n!}{{e^n}^2}$$ Then $$\lim_\limits{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=\lim_\limits{n\to\infty}\left|\frac{(n+1)!}{{e^{(n+1)}}^2}\cdot\frac{{e^n}^2}{n!}\right| $$ We can take away the absolute value bars and then simplify. $$ \lim_\limits{n\to\infty}\frac{(n+1)n!\cdot {e^n}^2}{e^{n^2+2n+1}n!}=\lim_\limits{n\to\infty}\frac{(n+1)}{e^{2n+1}}$$ so, as $n\to \infty$, $$\lim_\limits{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|\to0 $$ thus, the series is convergent by the Ratio Test.
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