Q. 24 - 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.

24. $\left\{1+\frac{1}{k}\right\}$

Step-by-Step Solution

Answer

The sequence is monotonic and bounded and the convergent.

The limit of the sequence $\left\{a_k\right\}=\left\{1+\frac{1}{k}\right\}$ is $1$.

Step 1. Given information

We have been given the sequence $\left\{1+\frac{1}{k}\right\}$.

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

$$\begin{gathered} \frac{1}{k}>\frac{1}{k+1} \\ 1+\frac{1}{k}>1+\frac{1}{k+1} \\ {a}_k>a_{k+1} \end{gathered}$$

The sequence is decreasing sequence and hence is monotonic.

The lower bound of the sequence $\left\{a_k\right\}=\left\{1+\frac{1}{k}\right\}$ is $0$.

Thus the sequence is bounded below.

The monotonic decreasing sequence with lower bound is convergent.

Step 3. Determine the limit of the sequence.

$$\begin{gathered} \underset{k\to \infty }{\lim }{a}_k \\ =\underset{k\to \infty }{\lim }\left(1+\frac{1}{k}\right) \\ =1+0 \\ =1 \end{gathered}$$

Thus, the limit of the sequence $\left\{a_k\right\}=\left\{1+\frac{1}{k}\right\}$ is $1$.