Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![**Problem 3**: Life is better when there are options. The series
\[
\sum_{n=3}^{\infty} \frac{1}{n(n-1)}
\]
can be analyzed in many different ways. You will discover them below!
(a) **Show this series a telescoping series.** (You may need to find a function \( f(x) \) so that \( S_n = f(n) \), where \(\{ S_n \}\) is the associated sequence of partial sums, see the hint in the next part.) Find the sum of this telescoping series.
(b) **Use the Integral Test to determine whether the series converges or diverges.** What is the value of the associated improper integral? Be sure to carefully write out your argument. (Hint: write out the telescoping series in the form \(\sum_{n=3}^{\infty} \frac{A}{n-1} - \frac{B}{n}\) for some constants \(A, B\) and apply the integral test to this series.)
(c) **Does the sum of the series in part (a) match the value of the improper integral from part (b)?** Should it? Briefly explain the reasoning that allowed you to make your conclusion. (A picture might be helpful too!)
(d) **Extra Credit**: Use the Limit Comparison Test to determine whether the series converges or diverges. Be sure to carefully write out your argument.](https://content.bartleby.com/qna-images/question/55bde63a-e93f-47aa-b4b4-6bf144a23089/f05e4e2e-42ae-4caa-9237-d24792c333e2/f30ar0p_processed.png)
Transcribed Image Text:**Problem 3**: Life is better when there are options. The series
\[
\sum_{n=3}^{\infty} \frac{1}{n(n-1)}
\]
can be analyzed in many different ways. You will discover them below!
(a) **Show this series a telescoping series.** (You may need to find a function \( f(x) \) so that \( S_n = f(n) \), where \(\{ S_n \}\) is the associated sequence of partial sums, see the hint in the next part.) Find the sum of this telescoping series.
(b) **Use the Integral Test to determine whether the series converges or diverges.** What is the value of the associated improper integral? Be sure to carefully write out your argument. (Hint: write out the telescoping series in the form \(\sum_{n=3}^{\infty} \frac{A}{n-1} - \frac{B}{n}\) for some constants \(A, B\) and apply the integral test to this series.)
(c) **Does the sum of the series in part (a) match the value of the improper integral from part (b)?** Should it? Briefly explain the reasoning that allowed you to make your conclusion. (A picture might be helpful too!)
(d) **Extra Credit**: Use the Limit Comparison Test to determine whether the series converges or diverges. Be sure to carefully write out your argument.
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- 1. In this problem, you'll look at the series > 1 )J: One can visualize this series as area, by drawing a k5/4 * k=10 box of area 103/4, a box of area 134 and so on. 115/4 (a) There are two ways to draw the series (as boxes) in question. Draw each of them together with the graph y= 34 on two different graphs. x5/4 (b) Use the picture to write an inequality comparing >) 1 to an improper integral. Check the k5/4 k=10 bounds of your improper integral carefully, and be sure you are comparing on the same domain. (c) Is the improper integral you wrote in (b) convergent or divergent? Explain. (d) Using the inequality you wrote in (b) and your answer to (c), what (if anything) can you conclude 1 :? about the convergence of 2 k5/4 k=10 (e) Repeat (b)-(d) with your second picture.arrow_forwardHello, Can someone show to solve problem 4 using the root test? Thank you!arrow_forwardI was looking for some help with this question, thanks.arrow_forward
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