Q. 28 - 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.

28. $\left\{k^{\frac{1}{10}}\right\}$

Step-by-Step Solution

Answer

The sequence is monotonic and bounded below and not convergent.

The sequence has no limit.

Step 1. Given information

We have been given the sequence $\left\{k^{\frac{1}{10}}\right\}$.

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

$a_k=k^{\frac{1}{10}}$

Now,

$$\begin{gathered} \frac{a_{k+1}}{a_k} \\ =\frac{{\left(k+1\right)}^{{\displaystyle \frac{1}{10}}}}{{\left(k\right)}^{{\displaystyle \frac{1}{10}}}} \\ ={\left(1+\frac{1}{k}\right)}^{\frac{1}{10}} \\ >1 \end{gathered}$$

So, $a_{k+!}>a_k$

The sequence is strictly increasing . The given sequence is monotonic.

The sequence is bounded below because 

$1\le a_k$ for k>0

The sequence has lower bound, therefore the sequence is bounded below.

The sequence is not convergent and has no limit.