Proving Convergence: Bounded Sequences and Absolute Convergence Test

[Bellarosa]

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.

Homework Equations

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.

 

[Dick]

A bounded sequence is not necessarily convergent.

 

[Bellarosa]

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

 

[Dick]

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...