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.

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

Short answer

The sequence is not monotonic and bounded and convergent.

The limit of he sequence is$0$.

Step-by-step solution

Step 1. Given information

We have been given the sequence$\left\{\frac{{\left(-1\right)}^k}{k+6}\right\}$.

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

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 as $k$increases.

The sequence $\left\{a_k\right\}=\left\{\frac{{\left(-1\right)}^k}{k+6}\right\}$is a bounded sequence because

$0\le k\le \frac{1}{7}$for $k>0$

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

Step 3. Determine the limit of the sequence.

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

The sequence is not monotonic but is bounded. Therefore, the sequence converges to$0$.