Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 11 - Infinite Sequences and Series - Exercises 11.7.1 - 11.7.9 - Page 764: 13


The series converges by the Ratio Test.

Work Step by Step

Use the Ratio Test:$\lim\limits_{n \to \infty}\Big|\frac{a_{n+1}}{a_n}\Big|$. Let $a_n=\frac{3^nn^2}{n!}$. Then, $\lim\limits_{n\to \infty}\Bigg|\frac{3^{n+1}(n+1)^2}{(n+1)!} \cdot \frac{n!}{3^nn^2}\Bigg|$. Because $n!$, $n^2$, and $3^n$ are positive for all $n>0$, we can remove the absolute value lines. Now, $\lim\limits_{n \to \infty}\Bigg[\frac{3^{n+1}(n+1)^2}{(n+1)!} \cdot \frac{n!}{3^nn^2}\Bigg]= \lim\limits_{n \to \infty}\Bigg[\frac{3(n+1)^2}{(n+1)n^2}\Bigg]=$ $3\cdot\lim\limits_{n \to \infty}\Bigg[\frac{n+1}{n^2}\Bigg]\to 0$ as $n\to \infty$ Therefore, the limit is $(3)(0)=0<1$, and the series converges by the Ratio Test.
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