Q. 20 Convergence or divergence of a s... [FREE SOLUTION]

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Chapter 7: Q. 20 (page 656)

Convergence or divergence of a series: For each of the series that follow, determine whether the series converges or diverges. Explain the criteria you are using and why your conclusion is valid.
${鈭�}_{k = 1}^{鈭�} \frac{\ln k}{k}$

Short Answer

Expert verified

By integral test, the given sequence diverges.

Step by step solution

Step 1. Given Information

The given sequence is ${鈭�}_{k = 1}^{鈭�} \frac{\ln k}{k}$ .

Step 2. Apply the integral test

According to the integral test, either ${鈭�}_{k = 1}^{鈭�} a_{k}$ and ${鈭�}_{1}^{鈭�} a_{k} d k$ both converges or both diverges.
In the given sequence, $a_{k} = \frac{\ln k}{k}$ .
Find the value of definite integral, $鈭� \frac{\ln k}{k} d k$ . Assume $\ln k = t$ . So, $\frac{1}{k} d k = d t$ .

$\begin{array}{rcl} 鈭� \frac{\ln k}{k} d k & = & 鈭� t d t \\ = & \frac{t^{2}}{2} + C \\ = & \frac{{(\ln k)}^{2}}{2} + C \end{array}$

Apply the limits and find the value of ${鈭�}_{1}^{鈭�} \frac{\ln k}{k} d k$ .

$\begin{array}{rcl} {鈭�}_{1}^{鈭�} \frac{\ln k}{k} d k & = & \frac{{(\ln 鈭�)}^{2}}{2} - \frac{{(\ln 1)}^{2}}{2} \\ = & 鈭� - 0 \\ = & 鈭� \end{array}$

Since ${鈭�}_{1}^{鈭�} a_{k} d k$ diverges, by integral test the given sequence diverges.

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91影视

Convergence or divergence of a series: For each of the series that follow, determine whether the series converges or diverges. Explain the criteria you are using and why your conclusion is valid.
${鈭�}_{k = 1}^{鈭�} \frac{\ln k}{k}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Apply the integral test

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91影视

Convergence or divergence of a series: For each of the series that follow, determine whether the series converges or diverges. Explain the criteria you are using and why your conclusion is valid. 鈭�k=1鈭�lnkk

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Apply the integral test

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Calculus

Discrete Mathematics

Statistics

Mechanics Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Convergence or divergence of a series: For each of the series that follow, determine whether the series converges or diverges. Explain the criteria you are using and why your conclusion is valid.
${鈭�}_{k = 1}^{鈭�} \frac{\ln k}{k}$