Chapter 7: Q. 19 (page 624)

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.
$\sum_{k = 1}^{\infty} \frac{k}{3 k + 100}$

Short Answer

Expert verified

The series diverges as $\lim_{k \to \infty} \frac{k}{3 k + 100} = \frac{1}{3}$ .

Step by step solution

Step 1. Given information.

Consider the given series,

$\sum_{k = 1}^{\infty} \frac{k}{3 k + 100}$

Step 2. Analyze the series.

The divergence test states that if the sequence $\{a_{k}\}$ does not converge to zero, then the series ${\sum a_{k}}_{k = 1}^{\infty}$ diverges.

The value of the sequence $\{a_{k}\} = \{\frac{1}{3 k + 100}\}$ is given below,

$\lim_{k \to \infty} a_{k} = \lim_{k \to \infty} (\frac{k}{3 k + 100}) = \lim_{k \to \infty} (\frac{1}{3 + \frac{100}{k}}) = \frac{1}{3 + 0} = \frac{1}{3}$

The series $\sum_{k = 1}^{\infty} \frac{k}{3 k + 100}$ by divergence test is divergent as $\lim_{k \to \infty} (\frac{k}{3 k + 100}) = \frac{1}{3}$ , which is non-zero.

Thus, the series $\sum_{k = 1}^{\infty} \frac{k}{3 k + 100}$ is divergent.

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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.
$\sum_{k = 1}^{\infty} \frac{k}{3 k + 100}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Analyze the series.

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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.∑k=1∞k3k+100

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Analyze the series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Theoretical and Mathematical Physics

Decision Maths

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Geometry

Pure Maths

Study anywhere. Anytime. Across all devices.

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.
$\sum_{k = 1}^{\infty} \frac{k}{3 k + 100}$