Q. 70 - Calculus

Question

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

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

Step-by-Step Solution

Answer

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

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 k increases.

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

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