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Essential Calculus: Early Transcendentals
Found in: Page 435
Essential Calculus: Early Transcendentals

Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

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Short Answer

\({\bf{37 - 40}}\) 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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Step by Step Solution


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\(\mathop {\lim }\limits_{n \to \infty } \left| {{a_n}} \right| = \mathop {\lim }\limits_{n \to \infty } n = \infty \).

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