Is the Series Alternating: (-1)^n / (arctan n)^n?

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Homework Help Overview

The discussion revolves around determining whether the series \((-1)^n / (arctan(n))^n\) is an alternating series. Participants are exploring the characteristics of the series in the context of convergence tests.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are questioning the definition of an alternating series and whether the series in question meets those criteria. There is a focus on the behavior of the terms as \(n\) approaches infinity and the implications for convergence tests.

Discussion Status

Some participants have offered insights regarding the behavior of the series terms, particularly in relation to their limits. There is an ongoing examination of whether the series is indeed alternating, with differing opinions on the limit of \(1/arctan(n)^n\) and its implications.

Contextual Notes

Participants are considering the behavior of \(arctan(n)\) as \(n\) increases, specifically its limit approaching \(\pi/2\), and how this affects the classification of the series. There is uncertainty about the convergence behavior of the series based on these observations.

remaan
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Homework Statement



Is this an alternating series : (-1)^n / ( arctan n ) ^ (n)

Homework Equations



I know that the alternating series should have (-1) to any power, but also the signs should be - , + , - , + ... or the opposit,

The Attempt at a Solution



I am ok with this I know that for testing Con. Or Div. of this we should use the root test

But this series. But the problem is " Is this seris alternating or not"

Becasue if not it will be treated normally with a root test and it will Con.

For me I think that it is not "Alternating "

What do you think ?
 
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Does 1/arctan(n)^n decrease to zero? That's what they want to you check.
 
mmm, do you mean the limit, if so no. It goes to inifity.
 
So, this is NOT an alternating sries, right ?
 
No. 1/arctan(n)^n doesn't go to infinity. Tell me why you think it does.
 
ohhh, Ya , I remembred it goes to Pie/ 2
Right ?
 
arctan(n) goes to pi/2. 1/arctan(n)^n doesn't.
 

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