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Chapter 7: Q. 51 (page 592)

In Exercises 51鈥�54 use the difference test in Theorem 7.6 to analyze the monotonicity of the given sequence.
$\{k^{2} - 5 k\}$

Short Answer

Expert verified

The given sequence is strictly increasing for $k 鈮� 2 .$

Step by step solution

01

Step 1. Given Information.

The given sequence is $\{k^{2} - 5 k\} .$

02

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 $a_{k} = k^{2} - 5 k .$

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

role="math" localid="1649229170271" $a_{k + 1} = {(k + 1)}^{2} - 5 (k + 1) a_{k + 1} = k^{2} + 2 k + 1 - 5 k - 5 a_{k + 1} = k^{2} - 3 k - 4$

According to the test,

role="math" localid="1649229456322" $a_{k + 1} - a_{k} 鈮� 0 k^{2} - 3 k - 4 - (k^{2} - 5 k) 鈮� 0 k^{2} - 3 k - 4 - k^{2} + 5 k 鈮� 0 2 k - 4 鈮� 0 k 鈮� 2$

Thus, the sequence is strictly increasing for $k 鈮� 2 .$

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Most popular questions from this chapter

Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.

${鈭�}_{k = 1}^{鈭�} 鈥� \frac{k^{2} + 1}{k!}$

Given that $a_{0} = - 3, a_{1} = 5, a_{2} = - 4, a_{3} = 2$ and ${鈭� a_{k}}_{k = 2}^{鈭�} = 7$ , find the value of ${鈭� a_{k}}_{k = 0}^{鈭�}$ .

Determine whether the series ${鈭�}_{n = 1}^{鈭�} \frac{{(- 1)}^{n + 1}}{3^{n}}$ converges or diverges. Give the sum of the convergent series.

Use either the divergence test or the integral test to determine whether the series in Given Exercises converge or diverge. Explain why the series meets the hypotheses of the test you select.

${鈭�}_{k = 1}^{鈭�} 鈥� \frac{1}{2} k^{鈭� 3 / 4}$

Use any convergence test from this section or the previous section to determine whether the series in Exercises 31鈥�48 converge or diverge. Explain how the series meets the hypotheses of the test you select.

${鈭�}_{k = 1}^{鈭�} \frac{1}{2 k + 7}$

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