Root and Ratio tests Inconclusive

In summary, the conversation discusses the task of finding examples of a divergent infinite series and a convergent infinite series of positive numbers with the properties that the limit of the ratio of consecutive terms and the limit of the nth root of the last term both equal 1. The divergent series "1+1+1+1+1+..." is given as an example, while the convergent series "1+1/2+1/3+1/4+..." is suggested as a hint. The conversation concludes with a confirmation that this series satisfies the desired properties.
  • #1
mattmns
1,128
6
Here is the question in the book:
--------------
Give an example of a divergent infinite series of positive numbers [itex]a_{n}[/itex] such that [itex]\lim_{n\rightarrow \infty}a_{n+1}/a_{n} = \lim_{n\rightarrow \infty}a_{n}^{1/n} = 1[/itex] and an example of a convergent infinite series of positive numbers with the same property.
-------------

For the divergent series I gave a somewhat trivial example. The series: 1+1+1+1+1+... which is certainly divergent and has the necessary limit properties.

For the convergent series though I am a little clueless. What should I be looking for. I know that the nth root of the "last" term should be 1, and also the "last" term should be approaching 0, but I just can't seem to get anything to satisfy both. Please don't post such a series, but instead if you could give some hints as to how I should go about finding one, and maybe other things that I should know when looking for such a sequence. Thanks!
 
Last edited:
Physics news on Phys.org
  • #2
hint: n^(1/n)-->1
 
  • #3
[itex]\sum_{n=0}^{\infty} 1/n^2[/itex] converges and has the properties. Thanks.

edit... Using your hint again, I could use [itex]\sum_{n=0}^{\infty} 1/n[/itex] as the divergent series with the desired properties which is probably a little nicer than my lame example :smile:
 
Last edited:
  • #4
Nice.

-----
 

1. What is the purpose of the root and ratio tests?

The root and ratio tests are used to determine the convergence or divergence of a series, based on the behavior of the terms in the series as n approaches infinity.

2. How do the root and ratio tests work?

The root test involves taking the nth root of the absolute value of each term in the series and then evaluating the limit as n approaches infinity. If the limit is less than 1, the series converges. The ratio test involves dividing each term in the series by the previous term and then evaluating the limit as n approaches infinity. If the limit is less than 1, the series converges.

3. What does it mean if the root and ratio tests are inconclusive?

If the root and ratio tests are inconclusive, it means that the tests were not able to determine whether the series converges or diverges. This could be due to the series being too complex or not meeting the requirements of the tests.

4. Can the root and ratio tests be used for all types of series?

No, the root and ratio tests are only applicable to series with positive terms. They cannot be used for alternating series or series with negative terms.

5. Are the root and ratio tests always accurate in determining convergence or divergence?

No, the root and ratio tests are not always accurate. It is possible for a series to pass the tests and still diverge, or fail the tests and still converge. They are just tools to help determine the behavior of a series, but further analysis may be needed to verify the results.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
172
  • Calculus and Beyond Homework Help
Replies
2
Views
704
  • Calculus and Beyond Homework Help
Replies
14
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
236
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
476
Replies
6
Views
652
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
775
Back
Top