Proof of Ratio Test on Infinite Series

In summary, the conversation discusses the possibility of proving the ratio test without using the geometric series. However, it is difficult to find such a proof since the motivation behind the ratio test is to use the comparison test, which requires knowledge of a convergent series. While it may be possible to prove the ratio test from the root test, this also relies on comparison with the geometric series. Several references are mentioned for those interested in seeing the proofs in action.
  • #1
Fooze
8
0

Homework Statement



I know that the ratio test can be proved using the geometric series knowledge, but I'm looking for a way to prove the ratio test without the geometric series at all.

The reason is that I'm proving the geometric series convergence with the ratio test, and my professor doesn't want me to use circular reasoning, because from what he remembered, he could only think of a proof with the geometric series.

Do you guys know of any proof that doesn't involve the geometric series?

I'm not trying to prove it myself, necessarily, just find the information somewhere. Any ideas?
 
Physics news on Phys.org
  • #2
I think you'll be hard pressed to find such a proof. The motivation behind the proof of the ratio test is to use the comparison test, which requires that we actually know a convergent series to begin with. Since it's easy to show the convergence of the geometric series using basic algebra and limits, there is no need to believe that any circular reasoning will result. I think you can prove the ratio test from the root test, but iirc, an easy proof of the root test also depends on comparison with the geometric series.
 
  • #3
Yes, that is what I was figuring... I was just hoping to prove my professor wrong. ;)

If anyone has any other thoughts though, please do let me know!
 
  • #4
snipez90 is exactly right. Here are some references if you wanted to see the proofs in action.

p. 190-193 of Mathematical Analysis by Tom Apostol
p. 60-67 of Principles of Mathematical Analysis by Walter Rudin
p. 161-162 of Advanced Calculus by Creighton Buck
 

1. What is the Proof of Ratio Test on Infinite Series?

The Proof of Ratio Test on Infinite Series is a mathematical method used to determine the convergence or divergence of an infinite series. It involves taking the ratio of consecutive terms in the series and analyzing the behavior of this ratio as the number of terms increases.

2. How is the Ratio Test different from other convergence tests?

The Ratio Test is different from other convergence tests because it focuses on the behavior of the ratio of consecutive terms in the series, rather than the individual terms themselves. This can be helpful in cases where the individual terms of a series do not have a clear pattern or do not approach zero.

3. What is the general formula for the Ratio Test?

The general formula for the Ratio Test is: lim n→∞ |an+1/an|. This means taking the limit as n approaches infinity of the absolute value of the ratio of the (n+1)th term to the nth term in the series.

4. How do you use the Ratio Test to determine convergence or divergence?

If the limit in the Ratio Test formula is less than 1, the series is absolutely convergent. If the limit is greater than 1 or infinite, the series is divergent. If the limit is exactly equal to 1, the test is inconclusive and another method may be needed to determine convergence or divergence.

5. Why is the Ratio Test useful in analyzing infinite series?

The Ratio Test is useful because it can be applied to a wide range of infinite series and can determine the convergence or divergence of a series even when other methods may not work. It also provides a clear and straightforward way to determine the behavior of a series as the number of terms increases.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
87
  • Calculus and Beyond Homework Help
Replies
2
Views
658
  • Calculus and Beyond Homework Help
Replies
14
Views
1K
  • Calculus and Beyond Homework Help
Replies
11
Views
2K
Replies
6
Views
525
  • Calculus and Beyond Homework Help
Replies
22
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
909
  • Calculus and Beyond Homework Help
Replies
29
Views
1K
Replies
13
Views
2K
  • Calculus and Beyond Homework Help
Replies
16
Views
3K
Back
Top