Question

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

$\left\{\frac{\cos k}{k}\right\}$

Short answer

The given sequence is a monotonic sequence and an unbounded sequence.

Step-by-step solution

Step 1. Given Information. 

The given sequence is$\left\{\frac{\cos k}{k}\right\}.$

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

To determine whether the sequences are monotonic or not, we will use the derivative test.

Let the function is $f\left(k\right)=\frac{\cos k}{k}.$

According to the derivative test,

$$\begin{gathered} f\text{'}\left(k\right)=\frac{k\left(-\sin k\right)-\cos k\left(1\right)}{k^2} \\ f\text{'}\left(k\right)=\frac{-k\sin k-\cos k}{k^2} \\ f\text{'}\left(k\right)=-\frac{k\sin k+\cos k}{k^2} \end{gathered}$$

Since $f\text{'}\left(k\right)<0.$

Therefore, the sequence is decreasing and it is a monotonic sequence.

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

We can depict that as the value ofkincreases, $a_k$decreases.

Thus, the sequence is unbounded.