Convergence/Divergence of Series: cos(1/n)

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Homework Help Overview

The discussion revolves around the convergence or divergence of the series involving the cosine function, specifically cos(1/n). Participants are exploring whether the series converges absolutely, converges conditionally, or diverges.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • The original poster considers using the limit comparison test or the ratio test as potential approaches. Another participant discusses evaluating the limit at infinity and the implications for convergence based on the behavior of the cosine function as n approaches infinity.

Discussion Status

Participants are actively engaging with the problem, with some suggesting methods for analysis and others providing insights into the behavior of the series. There is a recognition of the need to check conditions for convergence, particularly in the context of alternating series.

Contextual Notes

There is a mention of the requirement for the last term of an alternating series to converge to zero, which is part of the discussion on convergence criteria. The original poster expresses uncertainty about how to begin the analysis.

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Homework Statement



question is if this series converge absolutely, converge conditionally, or it diverges?

here it is: http://img442.imageshack.us/img442/5899/untitled8tn.jpg

Homework Equations



cos1/n

The Attempt at a Solution



not sure where to start, maybe with the limit comparison test or ratio test?
please help. thanks.
 
Last edited by a moderator:
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\sum_{n=1}^{\infinity} -1^n \cos \frac{1}{n}. First check if the limit at infinity is less than 1. If each term is more than 1, it won't converge. So as we take the limit, Cos 0, its equal to 1. Since its 1, it diverges. So it can't be absolutely convergent.

For an alternating series, the last term has to converge to zero as well, so its divergent as well.
 
Last edited:
thanks for your help.
 
No problemo :)
 

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