Question

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

A bounded and convergent sequence that is not eventually monotonic.

Short answer

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

Step-by-step solution

Step 1. Given information.

Consider the given question,

A bounded and 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 the general term is given below,

localid="1649179547266" $a_k=\left\{\frac{{\left(-1\right)}^k}{k+6}\right\}$

The sequence $\left\{a_k\right\}=\left\{\frac{{\left(-1\right)}^k}{k+6}\right\}$is not a monotonic sequence as 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 limit of the sequence.

The sequence $\left\{a_k\right\}=\left\{\frac{{\left(-1\right)}^k}{k+6}\right\}$is a bounded sequence as localid="1649179828821" $0\le a_k\le \frac{1}{7}$as localid="1649179831465" $x>0$.

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

Take the limit of the sequence,

localid="1649179740929" $$\begin{gathered} \underset{k\to \infty }{\lim }{a}_k=\underset{k\to \infty }{\lim }\left(\frac{{\left(-1\right)}^k}{k+6}\right) \\ =0 \end{gathered}$$

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

Hence, the sequence converges to $0$.