Q. 23 - Calculus

Question

For each of the sequences in Exercises 23–52 determine whether the sequence is monotonic or eventually monotonic and whether the sequence is bounded above and/or below. If the sequence converges, give the limit.

23. $\left\{11\right\}$

Step-by-Step Solution

Answer

The sequence is monotonic and bounded and the convergent.

The limit of the sequence $\left\{a_k\right\}=\left\{11\right\}$ is $11$.

Step 1. Given information

We have been given the sequence $\left\{11\right\}$.

Step 2. Determine whether the sequence is monotonic or eventually monotonic and whether the sequence is bounded above and/or below.

The constant sequence is always monotonic and bounded.

The monotonic and bounded sequence is always convergent.

The sequence $\left\{a_k\right\}=\left\{11\right\}$ is convergent.

Step 3. Determine the limit of the sequence.

$$\begin{gathered} \underset{k\to \infty }{\lim }{a}_k \\ =\underset{k\to \infty }{\lim }\left(11\right) \\ =11 \end{gathered}$$

Thus, the limit of the sequence $\left\{a_k\right\}=\left\{11\right\}$ is $11$.