Question

\({\bf{37 - 40}}\) Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?

\({{\bf{a}}_{\bf{n}}}{\bf{ = n( - 1}}{{\bf{)}}^{\bf{n}}}\)

Short answer

Neither monotonic nor bounded.

Step-by-step solution

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