Checking Answers for Comparison Test

In summary: This is because the comparison test doesn't work with series with LARGER members.This is incorrect. If a given series A, converges and you compare it against another series say B, in which each member of B is LESS than A then B may or may not be smaller than A. This is because the comparison test works with series with LESS members.This is correct.
  • #1
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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.)


A.) for n > 2, [tex]\frac{n}{n^3-4}[/tex] < [tex]\frac{2}{n^2}[/tex], and the series [tex]2\sum \frac{1}{n^2}[/tex] converges, so by the comparison test, the series [tex]\sum \frac{n}{n^3-4}[/tex] converges

B.) for n > 2, [tex]\frac{ln(n)}{n^2}[/tex] > [tex]\frac{1}{n^2}[/tex], and the series [tex]\sum \frac{1}{n^2}[/tex] converges, so by the comparison test, the series [tex]\sum \frac{ln(n)}{n^2}[/tex] converges

C.) for n > 2, [tex]\frac{1}{n^2-6}[/tex] < [tex]\frac{1}{n^2}[/tex], and the series [tex]\sum \frac{1}{n^2}[/tex] converges, so by the comparison test, the series [tex]\sum \frac{1}{n^2-6}[/tex] converges


For A.) and C.) sounds true, by the p-series, both converges so that means mean the original series converges, so A.) and C.) are True right?

and B.) I also think it's true by the p-series.
 
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  • #2
Certainly all the series converge (assuming they start at [itex]n \geq 1[/itex]). I want you to look much more closely at the arguments they present though, and see if you can find anything wrong with any of them.
 
  • #3
Data said:
sure you can, for example

[tex]\frac{1}{n^{\frac{3}{2}}} > \frac{\ln n}{n^2} \ \forall n > 0.[/tex]

Their particular argument is obviously wrong, though. If you look at their arguments for A and C, there are problems there too (what happens if [itex]n=3[/itex]? Do the inequalities they claim actually hold?).

Sorry, I pulled my reply because it seemed inappropriate after yours, which I did not know was going to be there. I didn't check their claimed inequality- a serious oversight on my part.
 
  • #4
Actually, before I deleted mine I fixed it a little - their argument for A is ok (originally I somehow read it as n^2 in the denominator, or something - but even that doesn't make sense, because I still thought it would converge! Strange, this brain of mine is).
 
  • #5
so A is false because there is a 2 in front of the 'sum of' sign(which you can't do)? and B and C are true?
 
  • #6
no. A is correct (of course putting the 2 in front doesn't matter! The sum itself is just a number, like any other). Look at our previous posts and then look carefully at the arguments for B and C. Why do you think they are correct?
 
  • #7
If a given series A, converges and you compare it against another series say B, in which each member of B is LARGER than A then nothing can be said for the series B.
 

1. What is the purpose of checking answers for the comparison test?

The purpose of checking answers for the comparison test is to ensure the accuracy of the test results and to identify any potential errors in the calculations or assumptions made during the test.

2. How do you check answers for the comparison test?

To check answers for the comparison test, you can first compare the given values to a known standard or reference point. Then, you can use mathematical equations or statistical methods to analyze the data and verify the results.

3. What are the common mistakes to look out for when checking answers for the comparison test?

Common mistakes to look out for when checking answers for the comparison test include incorrect data entry, calculation errors, and using the wrong formula or method for the given problem. It is important to double check all steps and calculations to ensure accuracy.

4. Can you use different methods to check answers for the comparison test?

Yes, there are various methods that can be used to check answers for the comparison test, such as graphical analysis, statistical tests, or using a different set of data for comparison. The method chosen will depend on the type of data and the specific goals of the test.

5. Are there any limitations to checking answers for the comparison test?

While checking answers for the comparison test can help validate the results, it is not foolproof and may not account for all possible scenarios or factors. It is important to consider the limitations of the test and to use multiple methods to verify the results for a more comprehensive analysis.

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