This is a well known theorem in the series of positive terms.
If the series Σan of positive monotone decreasing terms is to converge then we must have not only lim(an)=0 but also lim(n(an))=0 . However the condition lim(n(an))=0 is only necessary , not a sufficient one for these type of series. If lim(nan) does not tend to zero then definitely the series diverges but lim(nan)=0 does not necessarily implies anything as to the possible convergence of the series. in fact the Abel series Σ(1/nlogn) diverges though lim(nan)=0
Get a good book on series to revise all these theorems.