Chapter 7: Q. 53 (page 592)

In Exercises 51–54 use the difference test in Theorem 7.6 to analyze the monotonicity of the given sequence.
$\{\frac{k}{k + 2}\}$

Short Answer

Expert verified

The given sequence is strictly increasing for $k \geq 1 .$

Step by step solution

Step 1. Given Information.

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

Step 2. Use the difference test.

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

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

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

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

According to the test,

$a_{k + 1} - a_{k} = \frac{k + 1}{k + 3} - \frac{k}{k + 2} = \frac{(k + 1) (k + 2) - k (k + 3)}{(k + 3) (k + 2)} = \frac{k^{2} + k + 2 k + 2 - k^{2} - 3 k}{(k + 3) (k + 2)} = \frac{2}{(k + 3) (k + 2)}$

Now,role="math" localid="1649230532100" $a_{k + 1} - a_{k} \geq 0 for all k \geq 1 .$

Thus, the sequence is strictly increasing for $k \geq 1 .$

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In Exercises 51–54 use the difference test in Theorem 7.6 to analyze the monotonicity of the given sequence.
$\{\frac{k}{k + 2}\}$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Use the difference test.

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In Exercises 51–54 use the difference test in Theorem 7.6 to analyze the monotonicity of the given sequence.kk+2

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Use the difference test.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Applied Mathematics

Pure Maths

Mechanics Maths

Probability and Statistics

Statistics

Study anywhere. Anytime. Across all devices.

In Exercises 51–54 use the difference test in Theorem 7.6 to analyze the monotonicity of the given sequence.
$\{\frac{k}{k + 2}\}$