A question from Rudin (Principles) Chapter 3:
Let $a_n\geq0$ and $\sum a_n$ diverges. What can be said about convergence/divergence of $\sum\frac{a_n}{1+na_n}$?
This one is being recalcitrant. Given that $x>y$ implies $\frac{x}{1+nx}>\frac{y}{1+ny}$ and when $a_n=1/n\log n$ the sum in question diverges, it seems plausible that in general the sum will always diverge, but I can't get a proof out. If it does diverge, it does so pretty slowly as $\frac{a_n}{1+na_n}=\frac{1}{n}-\frac{1}{n+n^2a_n}\leq\frac{1}{n}.$