Chapter 7: Q. 25 (page 631)

In Exercises $21 - 30$ use one of the comparison tests to determine whether the series converges or diverges. Explain how the given series satisfies the hypotheses of the test you use.
$\sum_{k = 1}^{\infty} \frac{1 + \ln k}{k}$ .

Short Answer

Expert verified

The series $\sum_{k = 1}^{\infty} \frac{1 + \ln k}{k}$ is divergent.

Step by step solution

Step 1. Given information

$\sum_{k = 1}^{\infty} \frac{1 + \ln k}{k}$ .

Step 2. The comparison test states that for ∑k=1∞ ak and ∑k=1∞ bk be two series with positive terms then,

If $\lim_{k \to \infty} \frac{a_{k}}{b_{k}} = L$ , where $L$ is any positive real number, then either both converge or both diverge.
If $\lim_{k \to \infty} \frac{a_{k}}{b_{k}} = 0$ , and $\sum_{k = 1}^{\infty} b_{k}$ converges, then $\sum_{k = 1}^{\infty} a_{k}$ converges.
If $\lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \infty$ , and $\sum_{k = 1}^{\infty} b_{k}$ diverges then, $\sum_{k = 1}^{\infty} a_{k}$ diverges.

Step 3. The term of the series ∑k=1∞ 1+ln kk are positive.

Now find $\sum_{k = 1}^{\infty} b_{k}$ for the given series.

$\sum_{k = 1}^{\infty} b_{k} = \sum_{k = 1}^{\infty} \frac{1}{k}$

Next find the $\lim_{k \to \infty} \frac{a_{k}}{b_{k}}$ for the given series.

$\lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \lim_{k \to \infty} \frac{\frac{1 + \ln k}{k}}{\frac{1}{k}} = \lim_{k \to \infty} 1 + \ln k = 1$

Step 4. From the obtained values,

The value of $\lim_{k \to \infty} \frac{a_{k}}{b_{k}} = 1$ which is finite non zero number.

The value of $\sum_{k = 1}^{\infty} b_{k} = \frac{1}{k}$ is divergent by $p -$ series test.

Therefore, $\sum_{k = 1}^{\infty} a_{k}$ is divergent.

Th given series is divergent.

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In Exercises $21 - 30$ use one of the comparison tests to determine whether the series converges or diverges. Explain how the given series satisfies the hypotheses of the test you use.
$\sum_{k = 1}^{\infty} \frac{1 + \ln k}{k}$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. The comparison test states that for ∑k=1∞ ak and ∑k=1∞ bk be two series with positive terms then,

Step 3. The term of the series ∑k=1∞ 1+ln kk are positive.

Step 4. From the obtained values,

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In Exercises 21-30use one of the comparison tests to determine whether the series converges or diverges. Explain how the given series satisfies the hypotheses of the test you use.∑k=1∞1+lnkk.

Short Answer

Step by step solution

Step 1. Given information

Step 2. The comparison test states that for ∑k=1∞ ak and ∑k=1∞ bk be two series with positive terms then,

Step 3. The term of the series ∑k=1∞ 1+ln kk are positive.

Step 4. From the obtained values,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Logic and Functions

Discrete Mathematics

Calculus

Probability and Statistics

Statistics

Study anywhere. Anytime. Across all devices.

In Exercises $21 - 30$ use one of the comparison tests to determine whether the series converges or diverges. Explain how the given series satisfies the hypotheses of the test you use.
$\sum_{k = 1}^{\infty} \frac{1 + \ln k}{k}$ .