Chapter 7: Q. 66 (page 593)

Determine whether the sequences in Exercises 63–74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
$a_{k} = \frac{k^{2}}{k!}$

Short Answer

Expert verified

The given sequence is monotonic and unbounded.

Step by step solution

Step 1. Given Information.

The given sequence is $a_{k} = \frac{k^{2}}{k!} .$

Step 2. Determine whether the sequences are monotonic or not.

To determine whether the sequences are monotonic or not, we will use the ratio test.

Let the general term of the sequence is $a_{k} = \frac{k^{2}}{k!} .$

So, the $a_{k + 1}$ term is

$a_{k + 1} = \frac{{(k + 1)}^{2}}{(k + 1)!} .$

According to the ratio test,

$\frac{a_{k + 1}}{a_{k}} = \frac{\frac{{(k + 1)}^{2}}{(k + 1)!}}{\frac{k^{2}}{k!}} = \frac{k! {(k + 1)}^{2}}{k^{2} (k + 1)!} = \frac{k! (k + 1) (k + 1)}{k^{2} (k + 1) k!} = \frac{k + 1}{k^{2}}$

Now, $\frac{a_{k + 1}}{a_{k}} < 1 for k \geq 2 .$

Thus, the sequence is eventually decreasing for $k \geq 2,$ and it is a monotonic sequence.

Step 3. Determine whether the given sequence is bounded or unbounded.

As we have shown that the sequence is eventually decreasing for $k \geq 2 .$ Thus, it has an upper bound of 2but it doesn't have a lower bound, so it is an unbounded sequence.

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Determine whether the sequences in Exercises 63–74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
$a_{k} = \frac{k^{2}}{k!}$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Determine whether the sequences are monotonic or not.

Step 3. Determine whether the given sequence is bounded or unbounded.

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Determine whether the sequences in Exercises 63–74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded.ak=k2k!

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Determine whether the sequences are monotonic or not.

Step 3. Determine whether the given sequence is bounded or unbounded.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Probability and Statistics

Discrete Mathematics

Pure Maths

Mechanics Maths

Statistics

Study anywhere. Anytime. Across all devices.

Determine whether the sequences in Exercises 63–74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
$a_{k} = \frac{k^{2}}{k!}$