Question

(Second) Problem 3: [up to 7.5 pts] Harper is asked to determine if $\sum_{k=3}^{\infty} \ln(\frac{k+3}{k+5})$ converges or diverges. They provide the argument below. Since $\lim_{k \to \infty} \ln(\frac{k+3}{k+5}) = \ln(1) = 0$, I know that $\sum_{k=3}^{\infty} \ln(\frac{k+3}{k+5})$ converges. Determine if Morgan's argument is correct or incorrect. If it is incorrect, explain what Morgan's error is, then determine if the series converges or diverges correctly. Make sure that your solutions adhere to the Additional Guidelines for Sequences and Series listed on Page 3 of the coversheet. Are you submitting a revision for this problem? Yes No If "Yes", what was your score on the problem originally? Explanation/Work:

          (Second) Problem 3: [up to 7.5 pts] Harper is asked to determine if $\sum_{k=3}^{\infty} \ln(\frac{k+3}{k+5})$ converges or diverges. They provide the argument below.

Since $\lim_{k \to \infty} \ln(\frac{k+3}{k+5}) = \ln(1) = 0$, I know that $\sum_{k=3}^{\infty} \ln(\frac{k+3}{k+5})$ converges.

Determine if Morgan's argument is correct or incorrect. If it is incorrect, explain what Morgan's error is, then determine if the series converges or diverges correctly. Make sure that your solutions adhere to the Additional Guidelines for Sequences and Series listed on Page 3 of the coversheet.

Are you submitting a revision for this problem?  Yes  No

If "Yes", what was your score on the problem originally?

Explanation/Work:

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Added by Reginald H.

Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert Brent Burkett

02/02/2024

Step 1

The fact that the limit of the terms goes to 0 is a necessary but not sufficient condition for the convergence of a series. The limit test only tells us that if the limit is not 0, the series diverges. If the limit is 0, further tests are needed to determine Show more…

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(Second) Problem 3: [up to 7.5 pts] Harper is asked to determine if $\sum_{k=3}^{\infty} \ln(\frac{k+3}{k+5})$ converges or diverges. They provide the argument below. Since $\lim_{k \to \infty} \ln(\frac{k+3}{k+5}) = \ln(1) = 0$, I know that $\sum_{k=3}^{\infty} \ln(\frac{k+3}{k+5})$ converges. Determine if Morgan's argument is correct or incorrect. If it is incorrect, explain what Morgan's error is, then determine if the series converges or diverges correctly. Make sure that your solutions adhere to the Additional Guidelines for Sequences and Series listed on Page 3 of the coversheet. Are you submitting a revision for this problem? Yes No If "Yes", what was your score on the problem originally? Explanation/Work:

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Transcript

00:01 Okay, we want to integrate from 3 to infinity the natural log of x over x dx.

00:09 Okay, this problem isn't as hard as you might think, or maybe it's harder than i think, but here's how you got to approach it.

00:17 What you want to do is do this.

00:20 You want to integrate from 3 and then pick some letter like b, b, and we're going to take the limit as b goes to infinity of this integral, the natural log of x over x dx.

00:33 All right, to integrate this, i'm going to let u equal the natural log of x, du equals 1 over x dx, so x times du equals dx.

00:47 When i make all of my little substitutions, i have u on top, i have x on the bottom, and in place of dx, i put x times du, which is nice because now the x's are just gone.

01:03 When we integrate this, we get u squared over 2.

01:07 So i want to take the limit as b goes to 0 or infinity of the natural log of x squared all over 2 evaluated from 3 to b.

01:27 B.

01:28 All right.

01:29 When i evaluate that, so it's going to be the limit as b goes to infinity of the natural log of b squared over 2 minus the natural log of 3 squared over 2.

01:46 All right.

01:47 Now, as we let b go to infinity, as b goes to infinity, the natural log of b is just going to to keep getting bigger and bigger and bigger and bigger.

01:58 And then when you square it, it's going to get bigger.

02:02 This right here is just going to go to infinity...

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