Absolute convergence of series?

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Homework Help Overview

The discussion revolves around the absolute convergence of series, specifically examining the relationship between the absolute convergence of the series \(\sum a_n\) and the series \(\sum a_n^2\). Participants are exploring the implications of absolute convergence and the behavior of terms in the series.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants are considering the implications of absolute convergence and whether the properties of the terms \(a_n\) can be applied to \(a_n^2\). There is discussion about the behavior of \(a_n^2\) when \(a_n\) is small versus when it is greater than 1, and whether the original poster's initial thoughts can be utilized further.

Discussion Status

The discussion is ongoing, with participants raising questions about the validity of their approaches and exploring different perspectives on the problem. Some guidance has been offered regarding the behavior of terms, but there is no clear consensus on the next steps or a definitive approach yet.

Contextual Notes

There is uncertainty regarding the implications of \(a_n\) being greater than 1 and how that affects the convergence of the squared series. Participants are also navigating the constraints of the problem without providing a complete solution.

SMA_01
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The question is: Show that if \suman from n=1 to ∞ converges absolutely, then \suman2 from n=1 to converges absolutely.

I'm not sure which approach to take with this.

I am thinking that since Ʃan converges absolutely, |an| can be either -an or an and for Ʃan2, an can be either negative or positive. But I'm not sure where to go with this.

Any help is appreciated.

Thank you.
 
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You know that if a_n is small (=smaller than 1), then a_n^2 is even smaller. So the squared series is smaller than the original series.

Can you do something with this idea??
 
So you mean a_n^2 would be bounded from above for smaller values? But what if it's greater than 1?
 
SMA_01 said:
So you mean a_n^2 would be bounded from above for smaller values? But what if it's greater than 1?

Yes, if it's greater than 1 then there is a problem. Try to find a way to solve this.
 
Do you think I can use what I stated in my first post?
 
SMA_01 said:
Do you think I can use what I stated in my first post?

Uh, not really. Well,it's not wrong what you said, but it won't help you much further.
 

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