The asumptions are: an>0 (an is a sequence, where n is a subscript) and Sum(an) diverges

What can be said about Sum(an/(1+n*an))

I believe that this series can diverge and converge.

I found that if we let an=(1/n), which satisfies our assumptions, then the Sum(an/(1+n*an))=Sum(an/2)=(1/2)Sum(an) which diverges.

I cannot find an example of a diverding an series for which the Sum converges.