Q. 73 - Calculus

Question

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

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

Step-by-Step Solution

Answer

The given sequence is not monotonic and unbounded.

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 of k.

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 k increases, $a_k$ decreases as well as increases.

Thus, the sequence is unbounded.