Determine if series converges or diverges

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Homework Help Overview

The discussion revolves around determining the convergence or divergence of two infinite series: Ʃ cos²(n)/(n²+8) and Ʃ 5n/(n²+1) * cos(2πn). Participants are exploring the properties of these series and the implications of oscillating functions within them.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants are questioning the reasoning behind the initial assumption of divergence for both series. There is a discussion about the oscillatory nature of the cosine function and its impact on convergence. Some participants suggest using specific theorems to analyze the series rather than relying on intuition.

Discussion Status

The conversation is ongoing, with participants reconsidering their initial thoughts on the convergence of the series. There is an acknowledgment of the need for a more rigorous approach, and some guidance has been offered regarding the use of series tests.

Contextual Notes

Participants are discussing the implications of the cosine function's behavior and are encouraged to consider formal theorems related to series convergence. There is a mention of the need to clarify which series tests are familiar to the participants.

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Homework Statement


Ʃ cos^2(n)/(n^2+8)

Ʃ 5n/(n^2+1) * cos(2πn)


Homework Equations





The Attempt at a Solution


I think that both series diverge. Can anyone validate this or tell me if I'm wrong?
 
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Why do you think they diverge?
 
I'm changing my mind. First one diverges because the cosine will oscillate between -1 and 1. The second one converges because it will always go to 0?
 
Convergence and divergence of infinite series is a counterintuitive and complicated matter. Don't base you're conclusion of guesses and intuition. Use theorems instead. Do you know any? Which ones may be useful for those series?
 
turbokaz said:
I'm changing my mind. First one diverges because the cosine will oscillate between -1 and 1. The second one converges because it will always go to 0?

You have cos2n which will always be positive since it's squared. As a result, cos2n ≤ 1 and so cos2n/(n2 + 8) ≤ 1/(n2 + 8). From there it's not too difficult to show whether it converges or diverges using one of the series tests.

Before looking at the second one, which series tests are you familiar with?
 

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