Alternative examples, alternating series test

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SUMMARY

The discussion centers on the criteria for the Alternating Series Test, specifically the three requirements: the series must alternate, the terms must be non-increasing, and the limit of the terms must approach zero. Participants explore various examples, including the series ∑((-1)^2n)/n, which fails to alternate, and ∑sin(n)/n, which converges. The conversation highlights the importance of understanding the divergence test, particularly that if the limit of the terms does not equal zero, the series diverges. Additionally, the participants engage in creating examples that meet or violate these conditions.

PREREQUISITES
  • Understanding of the Alternating Series Test
  • Familiarity with convergence and divergence of series
  • Knowledge of limit concepts in calculus
  • Basic proficiency in mathematical notation and series representation
NEXT STEPS
  • Study the Divergence Test for series
  • Learn about the conditions for convergence in series
  • Explore examples of alternating series that converge and diverge
  • Investigate the implications of the limit of terms in series
USEFUL FOR

Students and educators in calculus, mathematicians interested in series convergence, and anyone seeking to deepen their 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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