Question

Determine whether the sequences in Exercises 63–74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded.

$\left\{{\left(-1\right)}^{k}k\right\}$

Short answer

The given sequence is not monotonic and unbounded.

Step-by-step solution

Step 1. Given Information. 

The given sequence is$\left\{{\left(-1\right)}^{k}k\right\}.$

Step 2. Determine whether the sequences are monotonic or not.

To determine whether the sequences are monotonic or not, we will find the terms of the sequence by putting the values ofk.

It is given${\left(-1\right)}^{k}k.$

So, let's find the terms,

$\left\{-1,2,-3,4,-5,...\right\}$

Since the sequence is neither increasing nor decreasing. Thus, the sequence is not monotonic.

Step 3. Determine whether the given sequence is bounded or unbounded.  

We can depict that as the value of kincreases, $a_k$decreases as well as increases.

Thus, the sequence is unbounded.