Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 11 - Infinite Series - 11.1 Sequences - Exercises - Page 538: 47

Answer

$\left\{ {{d_n}} \right\}$ diverges to infinity.

Work Step by Step

We have $\mathop {\lim }\limits_{n \to \infty } {d_n} = \mathop {\lim }\limits_{n \to \infty } \left( {\ln {5^n} - \ln n!} \right) = \mathop {\lim }\limits_{n \to \infty } \left( {n\ln 5} \right) - \mathop {\lim }\limits_{n \to \infty } \left( {\ln n!} \right)$, $\mathop {\lim }\limits_{n \to \infty } {d_n} = \left( {\ln 5} \right)\mathop {\lim }\limits_{n \to \infty } n - \mathop {\lim }\limits_{n \to \infty } \left( {\ln n!} \right)$. Either the sequence $\left\{ {\ln n!} \right\}$ converges or diverges, since $\left\{ n \right\}$ diverges to infinity, so $\left\{ {{d_n}} \right\}$ diverges to infinity.
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