Q. 30 - 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.

30. $\left\{{\left(\frac{1}{5}\right)}^k\right\}$

Step-by-Step Solution

Answer

The sequence is monotonic and bounded and convergent.

The limit of the sequence is $0$.

Step 1. Given information

We have been given the sequence $\left\{{\left(\frac{1}{5}\right)}^k\right\}$.

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

$a_k={\left(\frac{1}{5}\right)}^k$

$$\begin{gathered} \frac{a_{k+1}}{a_k}=\frac{{\left({\displaystyle \frac{1}{5}}\right)}^{k+1}}{{\left({\displaystyle \frac{1}{5}}\right)}^k} \\ ={\left(\frac{1}{5}\right)}^{k+1-k} \\ =\frac{1}{5} \\ <1 \end{gathered}$$

Thus, $a_{k+1}<a_k$

The sequence is strictly decreasing. The sequence is monotonic.

$a_k<1$ for $k>0$

Also

$0\le a_k<1$

The sequence is bounded.

The sequence is convergent.

Step 3. Determine the limit of the sequence.

$$\begin{gathered} \underset{k\to \infty }{\lim }{a}_k \\ =\underset{k\to \infty }{\lim }{\left(\frac{1}{5}\right)}^k \\ =0 \end{gathered}$$

Thus, the limit of the sequence $\left\{a_k\right\}=\left\{{\left(\frac{1}{5}\right)}^k\right\}$ is $0$.