Chapter 7: Q. 24 (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.
$\sum_{k = 1}^{\infty} \frac{\sqrt{k}}{1 + k^{2}}$ .

Short Answer

Expert verified

The series $\sum_{k = 1}^{\infty} \frac{\sqrt{k}}{1 + k^{2}}$ is convergent.

Step by step solution

Step 1. Given information

$\sum_{k = 1}^{\infty} \frac{\sqrt{k}}{1 + k^{2}}$ .

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 terms of the series ∑k=1∞ k1+k2 are positive.

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

$\sum_{k = 1}^{\infty} b_{k} = \sum_{k = 1}^{\infty} \frac{k^{\frac{1}{2}}}{k^{2}} = \sum_{k = 1}^{\infty} \frac{1}{k^{\frac{3}{2}}}$

Step 4. Next find limk→∞ akbk for the given series.

$\lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \lim_{k \to \infty} \frac{\frac{\sqrt{k}}{1 + k^{2}}}{\frac{1}{k^{\frac{3}{2}}}} = \lim_{k \to \infty} \frac{k^{\frac{3}{2}} \sqrt{k}}{1 + k^{2}} = \lim_{k \to \infty} \frac{k^{2}}{1 + k^{2}} = \lim_{k \to \infty} \frac{k^{2}}{k^{2} (1 + \frac{1}{k^{2}})} = \lim_{k \to \infty} \frac{1}{(1 + \frac{1}{k^{2}})} = 1$

Step 5. From the obtained values,

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

The value of role="math" localid="1649153106872" $\sum_{k = 1}^{\infty} b_{k} = \sum_{k = 1}^{\infty} \frac{1}{k^{\frac{3}{2}}}$ is convergent by using $p -$ series test.

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

Hence, the given series is convergent.

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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.
$\sum_{k = 1}^{\infty} \frac{\sqrt{k}}{1 + k^{2}}$ .

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 terms of the series ∑k=1∞ k1+k2 are positive.

Step 4. Next find limk→∞ akbk for the given series.

Step 5. 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∞k1+k2.

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 terms of the series ∑k=1∞ k1+k2 are positive.

Step 4. Next find limk→∞ akbk for the given series.

Step 5. 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

Statistics

Mechanics Maths

Probability and Statistics

Applied Mathematics

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.
$\sum_{k = 1}^{\infty} \frac{\sqrt{k}}{1 + k^{2}}$ .