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Expert-verified Found in: Page 435 ### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280 # $${\bf{37 - 40}}$$ Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?$${{\bf{a}}_{\bf{n}}}{\bf{ = n( - 1}}{{\bf{)}}^{\bf{n}}}$$

Neither monotonic nor bounded.

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## Definition

A sequence$$\left\{ {{a_n}} \right\}$$is called increasing if $${a_n} < {a_{n + 1}}$$for all, that is, $${a_1} < {a_2} < {a_3} < \cdots$$. It is called decreasing if $${a_n} > {a_{n + 1}}$$ for all. A sequence is monotonic if it is either increasing or decreasing.

## For increasing or decreasing & bound

Consider the sequence$${a_n} = n{( - 1)^n}$$.

The sequence is not monotonic because the terms are alternating.

Also, the sequence is not bounded because$$\mathop {\lim }\limits_{n \to \infty } \left| {{a_n}} \right| = \mathop {\lim }\limits_{n \to \infty } n = \infty$$. ### Want to see more solutions like these? 