Integral Comparison Test Problem

[jessicafanfan]

Hi there,

I ran into a problem while doing this question.
I double checked all my answers but still don't know where the problem lies. Can someone please point me to the right dierection?

Thank you so much!

 

[JeffM]

Hi there,

I ran into a problem while doing this question.
I double checked all my answers but still don't know where the problem lies. Can someone please point me to the right dierection?

Thank you so much!

It is impossible to show you where you are going wrong in your work if you don't show us your work.

 

[jessicafanfan]

My process

It is impossible to show you where you are going wrong in your work if you don't show us your work.

I first went through all the options by simplifying the equations into it dominant terms like 1/n^2-1 would be 1/n^2 and entered Correct or Incorrect straightaway since I know that for p-series 1/n^p, if p>1 it converges. I also double checked that the equations are smaller than the other or bigger than the other when I plotted it on desmos.

But I still don't know which one out of all of them is wrong...

Thanks!!

 

[JeffM]

I first went through all the options by simplifying the equations into it dominant terms like 1/n^2-1 would be 1/n^2 and entered Correct or Incorrect straightaway since I know that for p-series 1/n^p, if p>1 it converges. I also double checked that the equations are smaller than the other or bigger than the other when I plotted it on desmos.

But I still don't know which one out of all of them is wrong...

Thanks!!

This will be an unsatisfactory answer, but the attached picture is so small that I can barely read it. Consequently, I am unable to work out the problems myself to see whether I agree with your answer.

I do notice, however, that the question asks you to fill in I for "incorrect" if the argument provided is invalid even though the conclusion given is correct. Thus, there are two ways for you to have a wrong answer: you reached the wrong conclusion or you reached the correct conclusion but the given argument is invalid.

Perhaps others can see the details in your tiny jpeg, but for me at least you will need to write out the question so I can see it well enough to work on it.

 

[jessicafanfan]

questions

This will be an unsatisfactory answer, but the attached picture is so small that I can barely read it. Consequently, I am unable to work out the problems myself to see whether I agree with your answer.

I do notice, however, that the question asks you to fill in I for "incorrect" if the argument provided is invalid even though the conclusion given is correct. Thus, there are two ways for you to have a wrong answer: you reached the wrong conclusion or you reached the correct conclusion but the given argument is invalid.

Perhaps others can see the details in your tiny jpeg, but for me at least you will need to write out the question so I can see it well enough to work on it.

Hello,
Sorry about the small photos. I hope this is better?
I did try many times to get the right answer by switching C to I or I to C. However, I just don't know which one is actually wrong...

Thanks again!

 

[jessicafanfan]

This will be an unsatisfactory answer, but the attached picture is so small that I can barely read it. Consequently, I am unable to work out the problems myself to see whether I agree with your answer.

I do notice, however, that the question asks you to fill in I for "incorrect" if the argument provided is invalid even though the conclusion given is correct. Thus, there are two ways for you to have a wrong answer: you reached the wrong conclusion or you reached the correct conclusion but the given argument is invalid.

Perhaps others can see the details in your tiny jpeg, but for me at least you will need to write out the question so I can see it well enough to work on it.

Actually, here is the question!
Thank you so much!
(1 point) Each of the following statements is an attempt to show that a given series is convergent or divergent using the Comparison Test (NOT the Limit Comparison Test.) For each statement, enter C (for "correct") if the argument is valid, or enter I (for "incorrect") if any part of the argument is flawed. (Note: if the conclusion is true but the argument that led to it was wrong, you must enter I.)


1. For all

,

, and the series

converges, so by the Comparison Test, the series

converges.
2. For all

,

, and the series

converges, so by the Comparison Test, the series

converges.
3. For all

,

, and the series

converges, so by the Comparison Test, the series

converges.
4. For all

,

, and the series

converges, so by the Comparison Test, the series

converges.
5. For all

,

, and the series

converges, so by the Comparison Test, the series

converges.
6. For all

,

, and the series

diverges, so by the Comparison Test, the series

diverges.

 

[ksdhart2]

Actually, here is the question!
Thank you so much!
(1 point) Each of the following statements is an attempt to show that a given series is convergent or divergent using the Comparison Test (NOT the Limit Comparison Test.) For each statement, enter C (for "correct") if the argument is valid, or enter I (for "incorrect") if any part of the argument is flawed. (Note: if the conclusion is true but the argument that led to it was wrong, you must enter I.)

1. For all \(\displaystyle n>1, \: \dfrac{ln(n)}{n^2} < \dfrac{1}{n^{1.5}}\), and the series \(\displaystyle \displaystyle \sum_{n=1}^{\infty} \: \dfrac{1}{n^{1.5}}\) converges, so by the Comparison Test, the series \(\displaystyle \displaystyle \sum_{n=1}^{\infty} \: \dfrac{ln(n)}{n^{2}}\) converges.

2. For all \(\displaystyle n>1, \: \dfrac{n}{6-n^3} < \dfrac{1}{n^2}\), and the series \(\displaystyle \displaystyle \sum_{n=1}^{\infty} \: \dfrac{1}{n^{2}}\)converges, so by the Comparison Test, the series \(\displaystyle \displaystyle \sum_{n=1}^{\infty} \: \dfrac{n}{6-n^{3}}\) converges.

3. For all \(\displaystyle n>2, \: \dfrac{n}{n^3-5} < \dfrac{2}{n^2}\), and the series \(\displaystyle \displaystyle \sum_{n=2}^{\infty} \: \dfrac{2}{n^{2}}\) converges, so by the Comparison Test, the series \(\displaystyle \displaystyle \sum_{n=2}^{\infty} \: \dfrac{n}{n^3-5}\)converges.

4. For all \(\displaystyle n>2, \: \dfrac{1}{n^2-1} < \dfrac{1}{n^2}\), and the series \(\displaystyle \displaystyle \sum_{n=2}^{\infty} \: \dfrac{1}{n^{2}}\) converges, so by the Comparison Test, the series \(\displaystyle \displaystyle \sum_{n=2}^{\infty} \: \dfrac{1}{n^2-1}\)converges.

5. For all \(\displaystyle n>1, \: \dfrac{arctan(n)}{n^3} < \dfrac{\pi}{2n^3}\), and the series \(\displaystyle \displaystyle \dfrac{\pi}{2} \cdot\sum_{n=1}^{\infty} \: \dfrac{1}{n^3}\) converges, so by the Comparison Test, the series \(\displaystyle \displaystyle \sum_{n=1}^{\infty} \: \dfrac{arctan(n)}{n^3}\) converges.

6. For all \(\displaystyle n>2, \: \dfrac{ln(n)}{n} > \dfrac{1}{n}\), and the series \(\displaystyle \displaystyle \sum_{n=2}^{\infty} \: \dfrac{1}{n}\) diverges, so by the Comparison Test, the series \(\displaystyle \displaystyle \sum_{n=2}^{\infty} \: \dfrac{ln(n)}{n}\) diverges.

1. You said it was Correct. I agree with your answer.
2. You said it was Incorrect. I agree with your answer.
3. You said it was Incorrect. I agree with your answer.
4. You said it was Correct. I disagree with your answer.
5. You said it was Correct. I agree with your answer.
6. You said it was Correct. I agree with your answer.

I agree with your answers on all of them except problem 4. Please share with us your work and reasoning that led you to say that it was Correct. Please include all of your work, even the parts you know for sure are wrong. Thank you.