Q. 68 - Calculus

Question

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

$\left\{1,\frac{3}{4},\frac{8}{9},\frac{15}{16},\frac{24}{25},...\right\}$

Step-by-Step Solution

Answer

The given sequence is a monotonic sequence and it has a bounded below sequence.

Step 1. Given Information.

The given sequence is $\left\{1,\frac{3}{4},\frac{8}{9},\frac{15}{16},\frac{24}{25},...\right\}.$

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 $a_k=\frac{k^2-1}{k^2}.$

According to the derivative test,

$$\begin{gathered} a_k=\frac{k^2-1}{k^2} \\ {a}_k=1-\frac{1}{k^2} \\ a{\text{'}}_k=\frac{2}{k^3} \end{gathered}$$

Thus, the derivative of the function is positive for all k > 1.

Therefore, the sequence is increasing and it is monotonic.

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

The given sequence is increasing, thus its first term is a lower bound but there is no upper bound. Therefore, the sequence is bounded below.