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.

Abscissas
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Hey guys, this one is just for funnsies. So when dealing with an alternating series test, 3 requirements must be met, :
Alternating
u(sub n) ≥ u(sub n+1) for all n ≥ N, for some integer N
u(sub n) → 0 as n → ∞.

So I have been coming up with examples where of these are true, and one isnt. A little bit further, I am not sure if this makes sense, but is possible to find an example for each where like, one diverges and one converges, or does that not make sense?
 
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Come on, come up with some examples yourself first! It's not hard to find some counterexamples! Tell us what you've got already, and I'll help you fill in the parts you don't have.

(By the way, there are not always examples possible where one diverges and one converges. I can see one obvious condition where it wouldn't be the case).
 
Oh right! forgot about that, havnt slept in waaayy to long. Alright so i think the sum of ((-1)^2n)/n wouldn't alternate (rule 1) but seems to follow the rest, sum of cos(pi)/n I think would alternate and go to zero. I am thinking the one that your thinking of, which I've had a hunch on, is u(sub n) ≥ u(sub n+1) for all n ≥ N, for some integer N for diverging, but i can't really put it into words or make sense of it,its just a hunch , probably due to my lack of sleep
 
The one I'm thinking of is ##u_n \rightarrow 0##. If this is false, can the series ever converge?
 
Last edited:
Well, couldn't it converge to someone other than 0?
 
also, don't forget that u (Sub n) is in absolute values, sorry forgot to put that
 
Do you know the "divergence test"?
 
Right, if it doesn't go to zero then it diverges because of the infinite sums. forgot about that. Yeah so i guess for that one it is impossible
 
OK, so you have ##\sum 1/n## which does not alternate but does satisfy the other conditions. This series does not converge. Do you have one which does converge?
 
  • #10
∑sin(n)/n I think would converge
 
  • #11
never mind it alternates haha
 
  • #12
∑Sin(npi)/n right? because it would keep going to 0s and not alternating
 
  • #13
Abscissas said:
∑sin(n)/n I think would converge

Does it alternate? I'm not so sure? Alternating means it gets positive and negative at each turn. Does this do this?
 
  • #14
Abscissas said:
∑Sin(npi)/n right? because it would keep going to 0s and not alternating

You could probably write this easier.
 
  • #15
I just tried to plug it into my calculator and it denied me. My instincts tell me it doesnt. but I put down sin(3pi/4)= negative and sin(3pi/4)= still negative, and before pi its positive
 
  • #16
∑cos/n?
 
  • #17
Again, doesn't seem alternating to me...
 
  • #18
Hang on, would ∑1/n work? because the absolute value of the next term has to be greater then the last right?
 
  • #19
Here is one ##\sum_n 2^{-n}##.
 
  • #20
Abscissas said:
Hang on, would ∑1/n work? because the absolute value of the next term has to be greater then the last right?

No, the sequence needs to be decreasing. Every next term needs to have smaller absolute value.
 
  • #21
Ohhhh, ∑((-1)^2n)/n would work too right?
 
  • #22
That makes a lot more sense, and destroys my hunch about that rule not working
 
  • #23
Abscissas said:
Ohhhh, ∑((-1)^2n)/n would work too right?

OK, but that is just ##\sum 1/n##.
 
  • #24
Ohh okay, i see what you did there now. I feel dumb :p. Well hey, thanks for helping, its a lot of fun , and I feel like I am getting a better understanding of this test now. Would you like to help me further?
 
  • #25
Of course, what else would you like help on?
 
  • #26
Im thinking a divergence example for the decreasing rule where both others are true, I am trying to come up with something, ill post it once i have something respectable
 
  • #27
1 - (1 + 1) + 1/2 - (1/2 + 1/2) + 1/3 - (1/3 + 1/3) + ...
 
  • #28
Thats a really cool one, written it would be ∑n-(n+n) but wouldn't it violate the a (sub n)→ 0. I think it would go to negative infinity right? That was a cool one though and I think to fix that you could put it over n, but nvm then it gets smaller. Oh by the way, i totally couldn't come up with anything, I am assuming this was a clue to send me in the right direction, so I am going to play around with it a little
 
  • #29
Abscissas said:
Thats a really cool one, written it would be ∑n-(n+n)

Write it a bit differently (and use ##1/n## instead of ##n##).
 
  • #30
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)
 

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