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.

$\left\{k!\right\}$

Short answer

The given sequence is monotonic, bounded below and not convergent.

Step-by-step solution

Step 1. Given Information 

We are given the sequence $\left\{k!\right\}$and we need to find if the sequence is monotonic, bounded and the limit if it is convergent.

Step 2. Finding monotonic 

The general term is $a_k=k!$.

The ratio

$$\begin{gathered} \frac{a_{k+1}}{a_k} \\ =\frac{\left(k+1\right)!}{k!} \\ =\frac{\left(k+1\right)k!}{k!} \\ =\left(k+1\right) \\ >1 \\ \therefore a_{k+1}>a_k \end{gathered}$$

The sequence is strictly increasing so it is monotonic.

Step 3. Finding bounded  

The sequence is bounded below because $1\le a_k$. The increasing sequence has a lower bound and is $1$and no upper bound.

Therefore, the given sequence is bounded below.

Step 4. Finding the limit  

The monotonic increasing sequence has no upper bound.

Hence, it is not convergent.