Chapter 7: Q. 53 (page 604)

Determine whether the sequence converges or diverges. If the sequence converges, give the limit.
$\{(k!)^{1 / k}\}$

Short Answer

Expert verified

Ans: The sequence ${a_{k}} = \{(k!)^{1 / k}\}$ is divergent.

Step by step solution

Step 1. Given information.

given,

$\{(k!)^{1 / k}\}$

Step 2. The objective is to determine whether the sequence is convergent or divergent and to find the limit of the sequence if the sequence is convergent.

In the sequence ${a_{k}} = \{(k!)^{1 / k}\}$ the general term is $a_{k} = (k!)^{1 / k}$

The ratio $\frac{a_{k + 1}}{a_{k}}$ gives

role="math" localid="1649302149750" $\begin{aligned} \frac{a_{k + 1}}{a_{k}} = \frac{((k + 1)!)^{1 / k + 1}}{(k!)^{1 / k}} \\ > 1 (For k > 0) \\ Thus, a_{k + 1} > a_{k} \end{aligned}$

The sequence ${a_{k}} = \{(k!)^{1 / k}\}$ is strictly increasing. The given sequence is monotonic.

Step 3. The sequence {ak}=(k!)1/k is bounded below because

$0 < a_{k}$ for $k > 0$

The sequence is an increasing sequence and doesn't have any upper bound.

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

Step 4. The monotonic increasing sequence is bounded above is convergent.

The monotonic decreasing sequence ${a_{k}} = \{(k!)^{1 / k}\}$ is not bounded above and hence is not convergent. Therefore, the given sequence is divergent.

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Determine whether the sequence converges or diverges. If the sequence converges, give the limit.
$\{(k!)^{1 / k}\}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The objective is to determine whether the sequence is convergent or divergent and to find the limit of the sequence if the sequence is convergent.

Step 3. The sequence {ak}=(k!)1/k is bounded below because

Step 4. The monotonic increasing sequence is bounded above is convergent.

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Determine whether the sequence converges or diverges. If the sequence converges, give the limit. (k!)1/k

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The objective is to determine whether the sequence is convergent or divergent and to find the limit of the sequence if the sequence is convergent.

Step 3. The sequence {ak}=(k!)1/k is bounded below because

Step 4. The monotonic increasing sequence is bounded above is convergent.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

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Logic and Functions

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Determine whether the sequence converges or diverges. If the sequence converges, give the limit.
$\{(k!)^{1 / k}\}$