Product of two series diverges

[the_student]

Homework Statement

Find sequences an and bn such that: an>0 and an→0, Ʃ bn is bounded, but Ʃanbn diverges.

The Attempt at a Solution

The idea is that bn should be -1^n or -1^(n+1) and when multiplied by an the odd (larger) terms of the new sequence diverge and overpower the smaller terms. Every sequence an I have tried still ends up converging (for example 1/n → 0 and diverges but (-1^n)/n converges.

 

[Dick]

What exactly are you trying that doesn't work? 1/n does diverge. Doesn't that suggest something?

 

[the_student]

Yes, 1/n does diverge, however when multiplied by the bounded sequence (-1^n) you get (-1^n)/n which converges. It seems like I am missing the point, but I have focused on 1/n for some time and can't come up with how I can get the product to behave as 1/n does.

 

[the_student]

I also played with the series Ʃ(1/k^2 - 1/k) which diverges but couldn't figure out a way for it to be the product of required an and bn.

 

[Dick]

I also played with the series Ʃ(1/k^2 - 1/k) which diverges but couldn't figure out a way for it to be the product of required an and bn.

What happens if you multiply the bounded sequence (-1)^n by the convergent sequence (-1)^n/n?

 

[the_student]

That would work perfectly but the requirement an > 0 for all n isn't satisfied with that sequence.

 

[Dick]

That would work perfectly but the requirement an > 0 for all n isn't satisfied with that sequence.

Good point. You are paying attention. Suppose you define an=(1/n) if n is even and 1/n^2 if n is odd? Pick an to be not monotone.

 

[the_student]

Well, I do precisely need an to be not monotone so that it fails Abel's Test and the Alternating Series Test, but I am a little concerned about defining an the way you mentioned. It seems like it would work, but we have never defined any of our series that way even in the in class examples.

 

[Dick]

Well, I do precisely need an to be not monotone so that it fails Abel's Test and the Alternating Series Test, but I am a little concerned about defining an the way you mentioned. It seems like it would work, but we have never defined any of our series that way even in the in class examples.

Time to start a new trend then. Can you show it works?

 

[the_student]

Thanks for the help. I feel better about it now that I know I was mostly there except for basically how to write my thoughts down.