Proving Convergence: Bounded Sequences and Absolute Convergence Test

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Homework Help Overview

The discussion revolves around proving the convergence of the sum of the product of two sequences, specifically when one sequence is bounded and the other is convergent. The subject area includes concepts from real analysis, particularly related to sequences and series.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants explore the implications of a bounded sequence and its relationship to convergence. Questions arise about whether the product of a bounded sequence and a convergent sequence will also converge, with some participants suggesting the need for a proof rather than assumptions.

Discussion Status

The discussion is active, with participants questioning the assumptions about bounded sequences and convergence. There is a recognition that simply stating a conclusion is insufficient without proof, and suggestions for using a comparison test have been made.

Contextual Notes

There is a noted distinction between bounded sequences and convergent sequences, with some confusion regarding their properties. The participants are navigating the requirements for proving convergence in the context of the absolute convergence test.

Bellarosa
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1. Prove that if a sequence (bn) is bounded and the sum |(an)| going from n= 1 to infinity converges, then the sum of the product of sequences (an)(bn) converges.



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3. Given that the sequence bn is bounded it is convergent, and by the absolute convergence test if the sum of the absolute value of the sequence (an) converges then so does the sum of the sequence of (an), therefore the sum of the product of the sequences an and bn converges.
 
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A bounded sequence is not necessarily convergent.
 
Oh yes that's true. So can I say that the sum of the product of a bounded sequence times a convergent sequence will be convergent? I'm assuming that the bounded sequence bn times the convergent sequence an will make the sum of their product convergent
 
Bellarosa said:
Oh yes that's true. So can I say that the sum of the product of a bounded sequence times a convergent sequence will be convergent? I'm assuming that the bounded sequence bn times the convergent sequence an will make the sum of their product convergent

Sure you can 'say that'. Anybody can 'say that'. Aren't you supposed to 'prove that'? I would think about setting up a comparison test...
 

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