Question

Give examples of sequences satisfying the given conditions or explain why such an example cannot exist.

A convergent sequence that is not eventually monotonic.

Short answer

Examples of the sequences is $\left\{\frac{{\left(-1\right)}^k}{k+6}\right\}$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

A convergent sequence that is not eventually monotonic.

Step 2. Take the sequence -1kk+6.

Consider the sequence $\left\{\frac{{\left(-1\right)}^k}{k+6}\right\}$.

In the sequence $\left\{a_k\right\}=\left\{\frac{{\left(-1\right)}^k}{k+6}\right\}$the general term is given below,

$a_k=\left\{\frac{{\left(-1\right)}^k}{k+6}\right\}$

The sequence $\left\{a_k\right\}=\left\{\frac{{\left(-1\right)}^k}{k+6}\right\}$is not a monotonic sequence because sign of $\left\{\frac{{\left(-1\right)}^k}{k+6}\right\}$ varies alternately as k increases.

Hence, the given sequence is not a monotonic sequence.

Step 3. Take the sequence 

The sequence $\left\{a_k\right\}=\left\{\frac{{\left(-1\right)}^k}{k+6}\right\}$is a bounded sequence because $0\le a_k\le \frac{1}{7}$for $k>0$.

The given sequence has upper and lower bounds, then the sequence is bounded.

The limit of the sequence is $\left\{a_k\right\}=\left\{\frac{{\left(-1\right)}^k}{k+6}\right\}$,

localid="1649178846922" $$\begin{gathered} \underset{k\to \infty }{\lim }a=\underset{k\to \infty }{\lim }\left(\frac{{\left(-1\right)}^k}{k+6}\right) \\ =0 \end{gathered}$$

The sequence$\left\{a_k\right\}=\left\{\frac{{\left(-1\right)}^k}{k+6}\right\}$is not monotonic but bounded.

Hence, the sequence converges to$0$.