Alternative examples, alternating series test

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Discussion Overview

The discussion revolves around the alternating series test, specifically exploring examples that meet or violate its requirements. Participants are examining conditions for convergence and divergence of various series, with a focus on identifying examples that illustrate these concepts.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants outline the requirements for the alternating series test, including the need for terms to alternate, decrease, and approach zero.
  • One participant suggests that it might be possible to find examples where one series converges and another diverges, though this is questioned by others.
  • Several series are proposed, such as ∑((-1)^2n)/n, which does not alternate, and ∑sin(n)/n, with uncertainty about its alternating nature.
  • Participants discuss the implications of the divergence test, noting that if the terms do not approach zero, the series cannot converge.
  • There is a suggestion that the series ∑1/n could serve as an example that meets some conditions but does not alternate.
  • One participant proposes a complex series involving alternating signs and terms that may diverge, prompting further discussion about its behavior.
  • Another participant expresses confusion about how a series can diverge while still having terms that approach zero.

Areas of Agreement / Disagreement

Participants do not reach a consensus on specific examples that satisfy the conditions of the alternating series test, and there are multiple competing views regarding the nature of the proposed series and their convergence or divergence.

Contextual Notes

Some participants express uncertainty about the definitions and implications of the terms involved, particularly regarding the behavior of series as they approach limits. There are also discussions about the need for clarity in notation and the representation of series.

Who May Find This Useful

This discussion may be useful for students and enthusiasts of mathematics, particularly those interested in series convergence tests and the properties of alternating series.

  • #31
Abscissas said:
So when dealing with these, should I avoid simplifying? because i got ∑-1/n but before simplification i got ∑1/n-(1/n+1/n)

Sure, but write it even differently. Write it so that the odd term is ##1/n## and the even term is ##-2/n##.
 
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  • #32
OHHHHHHHHH I see what you did now ∑1/n-2/n which would diverge because to goes to 2 and your a genius haha thanks man
 
  • #33
Idk why, but i was trying to make it converge
 
  • #34
But wait, it would then violate the rule that it goes to 0, and i feel like I am wrong on this one, but its not alternating
 
  • #35
Because each sum, is negative, or when we say its alternating do we just mean each term?
 
  • #36
each term of the series*
 
  • #37
It means the terms converge to zero.
 
  • #38
So when i read this, I think of a wave dimming. Is this the right idea?
 
  • #39
Uh, I don't know. I don't have that picture in my mind. But yours could be helfpul.
 
  • #40
Okay, so here is my problem, I am working on 2(1/n)^n. The ^n is to make it alternate. the 1/n is to make it go to zero, and then the 2 is to make it not go to zero. How can something diverge, and still go to zero?
 
  • #41
2(-1/n)^n, sorry about that
 

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