Quick Question on Limit Comparison Test

Click For Summary

Discussion Overview

The discussion revolves around the applicability of the Limit Comparison Test (LCT) to alternating series, particularly in the context of the series \(\sum_{n=1}^{\infty} \frac{(-1)^n}{n}\). Participants explore the conditions under which the LCT can be used, the role of absolute values, and the relationship with the Alternating Series Test (AST).

Discussion Character

  • Debate/contested
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants question whether the LCT can be applied to alternating series, noting that standard conditions require the series terms to be positive.
  • Others suggest that comparisons can be made using the absolute values of the series, implying that this might allow the use of LCT on alternating series.
  • One participant points out the existence of the Alternating Series Test (AST) specifically for alternating series, questioning the necessity of using LCT in such cases.
  • A participant shares their attempt to apply LCT to the series \(\sum_{n=1}^{\infty} \frac{(-1)^n}{n}\) and expresses confusion over the results, indicating a divergence conclusion that seems incorrect.
  • Another participant emphasizes that the limit comparison test does not utilize absolute values, which raises concerns about the validity of comparisons involving oscillating terms.
  • Some participants agree that the LCT is generally applicable only to positive series, while others express uncertainty about the implications of using it with alternating series.
  • There is a mention of the convergence of the alternating harmonic series, with some participants recalling its value as \(-\log(2)\) and discussing its convergence properties.
  • One participant expresses frustration over mixed responses regarding the use of LCT on alternating series and the potential redundancy of LCT when AST is available.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the applicability of the LCT to alternating series, with multiple competing views presented regarding the conditions and methods of comparison.

Contextual Notes

There are unresolved mathematical steps and assumptions regarding the conditions under which LCT can be applied to alternating series, particularly concerning the use of absolute values and the nature of the comparison series.

Anonymous217
Messages
355
Reaction score
2
Can the limit comparison test work on alternating series? I looked at the conditions online and it said that the series a_n and b_n must be > 0 in order to use it. To be specific it said, "Suppose that a_n > 0 and b_n>0 for all n> or = to N (N a positive integer)."
I asked my teacher about this and he said that it does work, and yet... I don't know. Maybe I misinterpreted the conditions.
 
Physics news on Phys.org
When dealing with alternating series, the comparison can be carried out with absolute values of the series in question. The comparison series must have positive terms.
 
Isn't there an alternating series test for just such situations?
 
To be particular, I'm curious about being able to use the Limit Comparison test for
[tex]\sum_{n=1}^{\infty} \frac{(-1)^n}{n}[/tex]
I understand that it's easier to do with the AST, but I tried this with the LCT and it didn't turn out nice.

I compared it to 1/n.
[tex]\lim_{n\rightarrow \infty} |\frac{1}{n} * \frac{n}{(-1)^n}|[/tex]
[tex]\lim_{n\rightarrow \infty} |\frac{1}{(-1)^n}|[/tex]
[tex]= 1[/tex]
1/n diverges, so (-1)^n / n must diverge..? That's what I got using this test that is, and it's wrong. What did I do incorrectly with my math?
 
The limit comparison test doesn't use absolute values. Since limn→∞ 1/(-1)n doesn't exist/diverges, this comparison test doesn't help.
 
Also, the alternating harmonic series does converge... even if I forgot exactly why, or to what value... is it log 2?
 
It is log(2) and comes from the http://en.wikipedia.org/wiki/Mercator_series" .
 
Last edited by a moderator:
Bohrok said:
The limit comparison test doesn't use absolute values. Since limn→∞ 1/(-1)n doesn't exist/diverges, this comparison test doesn't help.
My first question asked if you could use the LCT on an alternating or negative series. Mathman replied that it's possible as long as you compare it to a positive function and while using absolute values. That's what I tried doing, and that's what my second post is about. Something seems off about the LCT or I made a mistake.

Char. Limit said:
Also, the alternating harmonic series does converge... even if I forgot exactly why, or to what value... is it log 2?
I'm aware that it does converge, because it should converge conditionally by using the AST. However, I'm wondering why the LCT doesn't work for this if apparently you are able to use it on a negative or alternating series. And it's -log(2), not log(2) by the way.
 
Last edited:
I also believe it will work with alternating series as long as the two series you're comparing alternate with the same sign so their limit converges to a number (since it doesn't use absolute values), unlike 1/(-1)n which oscillates between -1 and 1.
 
  • #10
The comparison tests can be applied only on the positive series.
 
  • #11
I'm getting mixed answers here.. :(

@Bohrok: But if I use an alternating series as a comparison for another alternating series, how will I know if the series I compared converges or diverges? And if you say AST, that makes the LCT useless because you could've done AST on the original series.

@Sweet GirL: I'm starting to believe in this. However, my teacher also said you can use it on alternating series by putting absolute values, but it doesn't seem like it works.
 
  • #12
My math teacher agrees with Sweet Girl. LCT can only be done on positive series.

Sorry, log(.5) then.
 
  • #13
Okay. I'll go challenge my math teacher. And by challenge, I mean ask how to do the above series using LCT and subtly hint that he's wrong.
 

Similar threads

MHB Limit Comparison Test: Does L Approaching Infinity Matter?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad What is the difference between these two power series theorems?
  • · Replies 5 ·
Replies
5
Views
3K
Graduate How Do Starting Points Affect Series Comparison Tests?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad How does the ratio test fail and the root test succeed here?
  • · Replies 6 ·
Replies
6
Views
4K
Undergrad Understanding the nth-term test for Divergence
  • · Replies 17 ·
Replies
17
Views
6K
Graduate Bounding Argument for Comparison Test
  • · Replies 3 ·
Replies
3
Views
1K
High School Continuation of sequence of supremums question
  • · Replies 9 ·
Replies
9
Views
2K
Undergrad Does the Limit Comparison Test Require an to Be Greater Than bn?
  • · Replies 2 ·
Replies
2
Views
2K
MHB Finding the interval of convergence for a series with lnn
  • · Replies 1 ·
Replies
1
Views
2K
MHB Finding if a series converges or diverges
  • · Replies 4 ·
Replies
4
Views
2K