Prove Convergence of Positive Series Squares

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Homework Help Overview

The discussion revolves around proving the convergence of the series defined by the squares of the terms of a convergent positive series. The original poster presents a series \(\Sigma a_{n}\) with \(a_{n} > 0\) and questions whether the convergence of this series implies the convergence of \(\Sigma a_{n}^{2}\).

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to reason that since the terms \(a_{n}\) approach zero, their squares \(a_{n}^{2}\) will also approach zero. They seek clarification on the correctness of this reasoning and how to formalize it.

Discussion Status

Some participants have suggested applying the ratio test as a potential approach. The original poster expresses a realization that the problem may be simpler than initially thought, indicating a shift in their understanding.

Contextual Notes

There is a separate inquiry about an arithmetic series that appears unrelated to the main discussion, which has prompted reminders about maintaining focus on the original topic.

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Homework Statement



The infinite series defined by \Sigma a_{n}, with a_{n}>0 are convergent. If then the series defined by \Sigma a_{n}^{2} coverges, prove it!

Homework Equations



The relevant equations has been stated above.

The Attempt at a Solution



Since every term in the first infinite series are positive the partial sums are monotone increasing. And, since it converges these will be bounded above. Then it feels like the series of the squares will be bounded above as well. Since, due to convergence, every term approaches zero.

Is it correct to say that since the term a_{n} tends to zero as n tends to infinity, its square also will?

Are my reasoning correct? How am I supposed to do it formally?

So very grateful for hints!
 
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Apply the ratio test.
 
Ohh... I was making it harder than it actually was!

Thank you so much! :)
 
find out the sum of arithematic series which has 25 terms and its middle number is 20
 
harryjose said:
find out the sum of arithematic series which has 25 terms and its middle number is 20
Does this have anything at all to do with the original question?

Please, please, please do not "hijack" someone else's thread to ask your own question! It is very easy to start your own thread.
 
I can only speak for myself and I can't see the connection to my thread!
 

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