Q 20. In Example 1 of Section 7.4 we u... [FREE SOLUTION]

Chapter 7: Q 20. (page 631)

In Example 1 of Section 7.4 we used the integral test to show that the series ${鈭�}_{k = 1}^{鈭�} \frac{k}{e^{k^{2}}}$ converges. Use the limit comparison test with the series ${鈭�}_{k = 1}^{鈭�} \frac{1}{e^{k}}$ to prove the same result.

Short Answer

Expert verified

$The value of \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} is \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 0 The series {鈭�}_{k = 1}^{鈭�} b_{k} = {鈭�}_{k = 1}^{鈭�} \frac{1}{e^{k}} is convergent . Therefore, the series {鈭�}_{k = 1}^{鈭�} a_{k} is also convergent . Hence, the series {鈭�}_{k = 1}^{鈭�} \frac{k}{e^{k^{2}}} is convergent .$

Step by step solution

Step 1. Given information is:

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

Step 2. Finding the term ∑ k=1∞ bk

$The terms of the series {鈭�}_{k = 1}^{鈭�} \frac{k}{e^{k^{2}}} are positive . The series {鈭�}_{k = 1}^{鈭�} b_{k} for the series {鈭�}_{k = 1}^{鈭�} \frac{k}{e^{k^{2}}} is given by : {鈭�}_{k = 1}^{鈭�} b_{k} = {鈭�}_{k = 1}^{鈭�} \frac{1}{e^{k}}$

Step 3. Evaluating limk→∞ akbk

$The ratio \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} is given by : \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = \lim_{k 鈫� 鈭�} \frac{\frac{k}{e^{k^{2}}}}{\frac{1}{e^{k}}} (Substitution) = \lim_{k 鈫� 鈭�} \frac{k}{e^{k}} (Simplify) = \lim_{k 鈫� 鈭�} \frac{1}{e^{k}} (Using L' Hopital' s rule) = 0$

Step 4. Result

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91影视

In Example 1 of Section 7.4 we used the integral test to show that the series ${鈭�}_{k = 1}^{鈭�} \frac{k}{e^{k^{2}}}$ converges. Use the limit comparison test with the series ${鈭�}_{k = 1}^{鈭�} \frac{1}{e^{k}}$ to prove the same result.

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Finding the term ∑ k=1∞ bk

Step 3. Evaluating limk→∞ akbk

Step 4. Result

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In Example 1 of Section 7.4 we used the integral test to show that the series鈭�k=1鈭�kek2 converges. Use the limit comparison test with the series鈭�k=1鈭�1ek to prove the same result.

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Finding the term ∑ k=1∞ bk

Step 3. Evaluating limk→∞ akbk

Step 4. Result

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Geometry

Applied Mathematics

Probability and Statistics

Mechanics Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In Example 1 of Section 7.4 we used the integral test to show that the series ${鈭�}_{k = 1}^{鈭�} \frac{k}{e^{k^{2}}}$ converges. Use the limit comparison test with the series ${鈭�}_{k = 1}^{鈭�} \frac{1}{e^{k}}$ to prove the same result.