Q. 28 In Exercises 21-30聽use one of t... [FREE SOLUTION]

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Chapter 7: Q. 28 (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.
${鈭�}_{k = 1}^{鈭�} \frac{1}{{(k - 蟺)}^{2}}$ .

Short Answer

Expert verified

The series ${鈭�}_{k = 1}^{鈭�} \frac{1}{{(k - 蟺)}^{2}}$ is convergent.

Step by step solution

Step 1. Given information

${鈭�}_{k = 1}^{鈭�} \frac{1}{{(k - 蟺)}^{2}}$ .

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

If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = L$ , where $L$ is any positive real number, then either both converge or both diverge.
If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 0$ , and ${鈭�}_{k = 1}^{鈭�} b_{k}$ converges, then ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges.
If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭�$ , and " width="9">鈭�k=1鈭�bkdiverges, then ${鈭�}_{k = 1}^{鈭�} a_{k}$ diverges.

Step 3. The terms of the series ∑k=1∞ 1k-π2 are positive.

Find ${鈭�}_{k = 1}^{鈭�} b_{k}$ for the given series.

${鈭�}_{k = 1}^{鈭�} b_{k} = {鈭�}_{k = 1}^{鈭�} \frac{1}{k^{2}}$

Next find $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}}$ for the given series.

$\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = \lim_{k 鈫� 鈭�} \frac{\frac{1}{{(k - 蟺)}^{2}}}{\frac{1}{k^{2}}} = \lim_{k 鈫� 鈭�} \frac{k^{2}}{{(k - 蟺)}^{2}} = \lim_{k 鈫� 鈭�} \frac{k^{2}}{k^{2} {(1 - \frac{蟺}{k})}^{2}} = \lim_{k 鈫� 鈭�} \frac{1}{{(1 - \frac{蟺}{k})}^{2}} = 1$

Step 4. From the obtained values,

The value of $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 1$ which is a finite positive number.

The value of localid="1649179701987" ${鈭�}_{k = 1}^{鈭�} b_{k} = {鈭�}_{k = 1}^{鈭�} \frac{1}{k^{2}}$ is convergent by $p -$ series test.

Therefore, the value of ${鈭�}_{k = 1}^{鈭�} b_{k}$ convergent.

Hence., the given series is convergent.

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91影视

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.
${鈭�}_{k = 1}^{鈭�} \frac{1}{{(k - 蟺)}^{2}}$ .

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 term then,

Step 3. The terms of the series ∑k=1∞ 1k-π2 are positive.

Step 4. From the obtained values,

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91影视

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.鈭�k=1鈭�1k-蟺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 term then,

Step 3. The terms of the series ∑k=1∞ 1k-π2 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

Geometry

Probability and Statistics

Discrete Mathematics

Decision Maths

Mechanics Maths

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.
${鈭�}_{k = 1}^{鈭�} \frac{1}{{(k - 蟺)}^{2}}$ .