Calculus (3rd Edition)

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

Chapter 11 - Infinite Series - Chapter Review Exercises - Page 591: 24


$$ a_n= n \pi $$ (other examples are possible.)

Work Step by Step

Consider $$ a_n= n \pi $$ then the series $\{a_n \}$ diverges. \begin{align*} \lim_{n\to \infty} a_n &= \infty \ne 0 \end{align*} However, for $\{\sin a_n \}$ \begin{align*} \lim_{n\to \infty} \sin a_n&= \lim_{n\to \infty} \sin n \pi \\ &=0 \end{align*} Hence, $\{\sin a_n \}$ converges.
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