Q. 64 - Calculus

Question

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

$\left\{\frac{1}{2k}\right\}$

Step-by-Step Solution

Answer

The given sequence is monotonic and unbounded. 

Step 1. Given Information.

The given sequence is $\left\{\frac{1}{2k}\right\}.$

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

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

Let the general term is $a_k=\frac{1}{2k}.$

So, the $a_{k+1}$ term is

$a_{k+1}=\frac{1}{2\left(k+1\right)}.$

According to the difference test,

$$\begin{gathered} a_{k+1}-a_k \\ =\frac{1}{2\left(k+1\right)}-\frac{1}{2k} \\ =\frac{1}{2k+2}-\frac{1}{2k} \\ =\frac{\left(2k\right)-\left(2k+2\right)}{\left(2k+2\right)\left(2k\right)} \\ =\frac{-2}{\left(2k+2\right)\left(2k\right)} \\ =\frac{-1}{\left(2k+2\right)\left(k\right)} \end{gathered}$$

Since, $a_{k+1}-a_k<0.$

Thus, the sequence is strictly decreasing and it is 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.

Thus, the sequence is unbounded.