The original poster is tasked with finding sequences \( a_n \) and \( b_n \) such that \( a_n > 0 \) and \( a_n \to 0 \), while the series \( \sum b_n \) is bounded, but the product series \( \sum a_n b_n \) diverges.
Participants are actively engaging with the problem, suggesting different sequences and questioning their properties. Some guidance has been offered regarding the definition of \( a_n \) to ensure it is not monotone, and there is acknowledgment of the original poster's progress in understanding the problem.
There is a concern about the requirement that \( a_n \) must be positive for all \( n \), and the discussion includes considerations of series that diverge or converge under specific conditions.
the_student said: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.
the_student said:That would work perfectly but the requirement an > 0 for all n isn't satisfied with that sequence.
the_student said: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.