Proving Whether an Alternating Series is Divergent or Convergent

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Homework Help Overview

The discussion revolves around determining the convergence or divergence of a specific sequence defined as an={1, 0, -1, 0, 1, 0, -1, 0, 1, ...}. Participants are exploring the nature of this sequence and its representation as a function.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • The original poster attempts to express the sequence using a function, suggesting an equation involving cosine. They question whether this representation is adequate or if a quadratic form would be better. Other participants inquire about the requirements for proving convergence, including the use of epsilon definitions.

Discussion Status

Participants are actively discussing the characteristics of the sequence, with some noting that it has multiple limit points, which suggests divergence. There is a recognition that the general term does not converge to zero, indicating divergence of the corresponding series. However, no consensus has been reached on the best approach to prove these points.

Contextual Notes

Some participants mention that they are not required to use epsilon definitions in their proofs, which may influence their approaches to demonstrating convergence or divergence.

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Homework Statement




Determine an explicit function for this sequence and determine whether it is convergent.

an={1, 0, -1, 0, 1, 0, -1, 0, 1, ...}


The Attempt at a Solution



I came up with this function:

an = cos(nπ/2), and wrote that as sigma notation from n=0 to infinity. Is that good or should I use a quadratic?

I don't think it converges because it just oscillates and never changes, but how do I prove that?
 
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Do you have to prove it using epsilon's and stuff??
If so, what is the definition of convergence? What is the negation?
 
see answer 5
 
Last edited:
I don't have to use epsilons. But I have to provide an equation showing it. And is the function I made satisfactory, or should I express it as a quadratic?
 
This sequence has 3 different limit points -1,0 and 1, therefore it does not have a limit.
 
If you reffer to the corresponding series,then it diverges because the general term does not converge to 0.
 

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