Alternative examples, alternating series test

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The discussion centers on the requirements for the alternating series test, specifically the conditions that must be met for convergence or divergence. Participants explore examples of series that either meet or fail these conditions, such as the series involving sine and cosine functions. There is a focus on understanding how to create examples where one series converges while another diverges, with some confusion about the definitions of alternating series and convergence. The conversation highlights the importance of the terms approaching zero and the behavior of the series as a whole. Overall, the discussion aims to deepen understanding of the alternating series test and its applications.
  • #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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