Question

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

$\left\{\frac{\cos \left(2\mathrm{\pi }k\right)}{k}\right\}$

Short answer

The given sequence is not a monotonic sequence but it is a bounded sequence.

Step-by-step solution

Step 1. Given Information. 

The given sequence is$\left\{\frac{\cos \left(2\mathrm{\pi }k\right)}{k}\right\}.$

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

The given sequence is not a monotonic sequence because the sign of $\frac{\cos \left(2\mathrm{\pi k}\right)}{k}$ differs as kincreases.

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

The given sequence is bounded because$-1\le \cos \left(2\mathrm{\pi k}\right)\le 1,\frac{-1}{k}\le \frac{\cos \left(2\mathrm{\pi k}\right)}{k}\le \frac{1}{k}.$