It seems to me that if then
which diverges, or have I misunderstood something?
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.
By the way, thank you for your interest and time. The reason why I believe it may converge is because the series in question is half the harmonic
mean of the series a_n and the series 1/n, both of which are divergent.
Any mean (including the harmonic mean) should be expected to be
between the two numbers, which means that the series in question (or
rather twice the series in question) should be bigger than either a_n
or 1/n, whichever is smaller.
The trouble is that you can't tell whether a_n will be bigger or 1/n,
and it might differ for different n. For example, if it was always
a_n, or if it was always 1/n, then it would be easy to prove that the
sum will always diverge if an does. But what if it changes? Maybe a_n
is smaller only sometimes. In fact, maybe a_n is very small (say
1/n^2) for most values of n, but rather large (say 1) for occasional
rare values of n (say perfect squares). Then a_n still diverges, but
the new series is very small for most values, and not quite as large
as a_n for the rare values, perhaps still small enough to converge.
I hope this provides the example you seek.
Consider the following sequence
then the series: diverges.
which does converge.
Now all that remains to provide your example is to replace the
zero terms in the equence with terms from
a convergent series of positive terms.