Chapter 7: Q. 55 (page 592)

In Exercises 55– 58 use the ratio test in Theorem 7.6 to analyze the monotonicity of the given sequence.
$\{\frac{k^{2}}{k!}\}$

Short Answer

Expert verified

The given sequence is eventually decreasing for $k \geq 2 .$

Step by step solution

Step 1. Given Information.

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

Step 2. Use the ratio test.

To analyze the monotonicity of the given sequence we will use the ratio test.

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

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

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

According to the ratio test,

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

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

Thus, the given sequence is eventually decreasing for $k \geq 2 .$

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In Exercises 55– 58 use the ratio test in Theorem 7.6 to analyze the monotonicity of the given sequence.
$\{\frac{k^{2}}{k!}\}$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Use the ratio test.

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In Exercises 55– 58 use the ratio test in Theorem 7.6 to analyze the monotonicity of the given sequence.k2k!

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Use the ratio test.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Mechanics Maths

Calculus

Geometry

Logic and Functions

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

In Exercises 55– 58 use the ratio test in Theorem 7.6 to analyze the monotonicity of the given sequence.
$\{\frac{k^{2}}{k!}\}$